Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^25 - 2*x^24 - x^23 + 13*x^22 + 50*x^21 + 3*x^20 - 29*x^19 - 5*x^18 + 128*x^17 - 146*x^16 - 239*x^15 + 32*x^14 + 747*x^13 - 262*x^12 - 1333*x^11 - 456*x^10 + 1762*x^9 + 962*x^8 - 1685*x^7 - 1308*x^6 + 1439*x^5 + 1916*x^4 - 95*x^3 - 410*x^2 + 94*x - 1)
gp: K = bnfinit(y^25 - 2*y^24 - y^23 + 13*y^22 + 50*y^21 + 3*y^20 - 29*y^19 - 5*y^18 + 128*y^17 - 146*y^16 - 239*y^15 + 32*y^14 + 747*y^13 - 262*y^12 - 1333*y^11 - 456*y^10 + 1762*y^9 + 962*y^8 - 1685*y^7 - 1308*y^6 + 1439*y^5 + 1916*y^4 - 95*y^3 - 410*y^2 + 94*y - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^25 - 2*x^24 - x^23 + 13*x^22 + 50*x^21 + 3*x^20 - 29*x^19 - 5*x^18 + 128*x^17 - 146*x^16 - 239*x^15 + 32*x^14 + 747*x^13 - 262*x^12 - 1333*x^11 - 456*x^10 + 1762*x^9 + 962*x^8 - 1685*x^7 - 1308*x^6 + 1439*x^5 + 1916*x^4 - 95*x^3 - 410*x^2 + 94*x - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^25 - 2*x^24 - x^23 + 13*x^22 + 50*x^21 + 3*x^20 - 29*x^19 - 5*x^18 + 128*x^17 - 146*x^16 - 239*x^15 + 32*x^14 + 747*x^13 - 262*x^12 - 1333*x^11 - 456*x^10 + 1762*x^9 + 962*x^8 - 1685*x^7 - 1308*x^6 + 1439*x^5 + 1916*x^4 - 95*x^3 - 410*x^2 + 94*x - 1)
\( x^{25} - 2 x^{24} - x^{23} + 13 x^{22} + 50 x^{21} + 3 x^{20} - 29 x^{19} - 5 x^{18} + 128 x^{17} + \cdots - 1 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $25$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[1, 12]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(42581619494519898305269398418425099361\)
\(\medspace = 1367^{12}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(32.00\) | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(OK))^(1/Degree(K));
oscar: (1.0 * dK)^(1/degree(K))
| |
| Galois root discriminant: | $1367^{1/2}\approx 36.9729630946723$ | ||
| Ramified primes: |
\(1367\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{5}a^{18}+\frac{1}{5}a^{17}+\frac{2}{5}a^{16}+\frac{2}{5}a^{15}+\frac{1}{5}a^{13}-\frac{1}{5}a^{12}-\frac{1}{5}a^{11}-\frac{1}{5}a^{10}+\frac{2}{5}a^{8}-\frac{1}{5}a^{7}-\frac{2}{5}a^{6}-\frac{1}{5}a^{5}+\frac{2}{5}a^{3}-\frac{1}{5}a^{2}-\frac{1}{5}a-\frac{2}{5}$, $\frac{1}{5}a^{19}+\frac{1}{5}a^{17}-\frac{2}{5}a^{15}+\frac{1}{5}a^{14}-\frac{2}{5}a^{13}+\frac{1}{5}a^{10}+\frac{2}{5}a^{9}+\frac{2}{5}a^{8}-\frac{1}{5}a^{7}+\frac{1}{5}a^{6}+\frac{1}{5}a^{5}+\frac{2}{5}a^{4}+\frac{2}{5}a^{3}-\frac{1}{5}a+\frac{2}{5}$, $\frac{1}{35}a^{20}+\frac{3}{35}a^{18}+\frac{17}{35}a^{17}-\frac{8}{35}a^{16}-\frac{3}{7}a^{15}-\frac{17}{35}a^{14}-\frac{13}{35}a^{13}-\frac{17}{35}a^{12}-\frac{16}{35}a^{11}-\frac{8}{35}a^{9}+\frac{3}{35}a^{8}+\frac{2}{5}a^{7}-\frac{8}{35}a^{6}-\frac{2}{7}a^{5}-\frac{8}{35}a^{4}+\frac{9}{35}a^{3}+\frac{1}{5}a^{2}-\frac{1}{7}a+\frac{6}{35}$, $\frac{1}{5005}a^{21}-\frac{31}{5005}a^{20}-\frac{459}{5005}a^{19}-\frac{62}{5005}a^{18}+\frac{557}{5005}a^{17}-\frac{199}{455}a^{16}-\frac{31}{143}a^{15}+\frac{122}{5005}a^{14}+\frac{519}{5005}a^{13}+\frac{296}{715}a^{12}+\frac{1707}{5005}a^{11}-\frac{694}{5005}a^{10}-\frac{1548}{5005}a^{9}+\frac{239}{1001}a^{8}+\frac{1371}{5005}a^{7}+\frac{69}{715}a^{6}-\frac{1504}{5005}a^{5}-\frac{877}{5005}a^{4}-\frac{1203}{5005}a^{3}-\frac{72}{385}a^{2}-\frac{263}{715}a+\frac{2432}{5005}$, $\frac{1}{5005}a^{22}+\frac{2}{1001}a^{20}-\frac{277}{5005}a^{19}-\frac{6}{385}a^{18}+\frac{349}{5005}a^{17}-\frac{261}{1001}a^{16}+\frac{1093}{5005}a^{15}-\frac{18}{91}a^{14}-\frac{2}{7}a^{13}-\frac{59}{715}a^{12}+\frac{2316}{5005}a^{11}-\frac{16}{77}a^{10}-\frac{5}{143}a^{9}-\frac{337}{5005}a^{8}+\frac{1943}{5005}a^{7}+\frac{2029}{5005}a^{6}+\frac{261}{5005}a^{5}-\frac{256}{715}a^{4}+\frac{1668}{5005}a^{3}+\frac{2176}{5005}a^{2}+\frac{5}{11}a-\frac{302}{715}$, $\frac{1}{95994173275}a^{23}-\frac{183483}{13713453325}a^{22}-\frac{5819479}{95994173275}a^{21}+\frac{905452656}{95994173275}a^{20}+\frac{9271386139}{95994173275}a^{19}+\frac{635793987}{19198834655}a^{18}+\frac{8474931123}{95994173275}a^{17}+\frac{47097051713}{95994173275}a^{16}-\frac{217167058}{3839766931}a^{15}+\frac{9947501383}{95994173275}a^{14}+\frac{208742447}{5646716075}a^{13}+\frac{803658604}{2742690665}a^{12}-\frac{45794007676}{95994173275}a^{11}-\frac{4130023323}{8726743025}a^{10}-\frac{28155216319}{95994173275}a^{9}+\frac{29267437406}{95994173275}a^{8}-\frac{29440808464}{95994173275}a^{7}+\frac{3835072528}{13713453325}a^{6}-\frac{16588743111}{95994173275}a^{5}+\frac{28724566809}{95994173275}a^{4}-\frac{3659491006}{19198834655}a^{3}+\frac{4492599453}{95994173275}a^{2}-\frac{22691376417}{95994173275}a-\frac{4984496639}{13713453325}$, $\frac{1}{43\!\cdots\!75}a^{24}+\frac{4087554299}{43\!\cdots\!75}a^{23}-\frac{13\!\cdots\!17}{62\!\cdots\!25}a^{22}+\frac{72\!\cdots\!41}{43\!\cdots\!75}a^{21}-\frac{24\!\cdots\!86}{43\!\cdots\!75}a^{20}-\frac{35\!\cdots\!72}{87\!\cdots\!35}a^{19}+\frac{11\!\cdots\!89}{25\!\cdots\!75}a^{18}-\frac{44\!\cdots\!97}{43\!\cdots\!75}a^{17}-\frac{76\!\cdots\!12}{17\!\cdots\!15}a^{16}+\frac{74\!\cdots\!89}{62\!\cdots\!25}a^{15}-\frac{71\!\cdots\!57}{33\!\cdots\!75}a^{14}+\frac{84\!\cdots\!42}{67\!\cdots\!95}a^{13}+\frac{10\!\cdots\!42}{62\!\cdots\!25}a^{12}-\frac{50\!\cdots\!34}{25\!\cdots\!75}a^{11}+\frac{79\!\cdots\!36}{43\!\cdots\!75}a^{10}+\frac{79\!\cdots\!63}{62\!\cdots\!25}a^{9}-\frac{60\!\cdots\!33}{33\!\cdots\!75}a^{8}-\frac{91\!\cdots\!66}{36\!\cdots\!25}a^{7}-\frac{11\!\cdots\!61}{43\!\cdots\!75}a^{6}-\frac{53\!\cdots\!61}{43\!\cdots\!75}a^{5}-\frac{30\!\cdots\!26}{87\!\cdots\!35}a^{4}-\frac{86\!\cdots\!57}{43\!\cdots\!75}a^{3}+\frac{13\!\cdots\!13}{43\!\cdots\!75}a^{2}-\frac{29\!\cdots\!99}{62\!\cdots\!25}a-\frac{66\!\cdots\!87}{87\!\cdots\!35}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $5$ |
Class group and class number
$C_{2}\times C_{2}$, which has order $4$ (assuming GRH)
sage: K.class_group().invariants()
gp: K.clgp
magma: ClassGroup(K);
oscar: class_group(K)
Unit group
sage: UK = K.unit_group()
magma: UK, fUK := UnitGroup(K);
oscar: UK, fUK = unit_group(OK)
| Rank: | $12$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( -1 \)
(order $2$)
| sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
oscar: torsion_units_generator(OK)
| |
| Fundamental units: |
$\frac{71\!\cdots\!37}{43\!\cdots\!75}a^{24}-\frac{13\!\cdots\!31}{43\!\cdots\!75}a^{23}-\frac{10\!\cdots\!12}{62\!\cdots\!25}a^{22}+\frac{87\!\cdots\!83}{43\!\cdots\!75}a^{21}+\frac{36\!\cdots\!84}{43\!\cdots\!75}a^{20}+\frac{84\!\cdots\!09}{43\!\cdots\!75}a^{19}-\frac{85\!\cdots\!92}{25\!\cdots\!75}a^{18}-\frac{10\!\cdots\!56}{39\!\cdots\!25}a^{17}+\frac{17\!\cdots\!97}{85\!\cdots\!75}a^{16}-\frac{99\!\cdots\!94}{48\!\cdots\!25}a^{15}-\frac{16\!\cdots\!64}{43\!\cdots\!75}a^{14}-\frac{36\!\cdots\!71}{43\!\cdots\!75}a^{13}+\frac{76\!\cdots\!09}{62\!\cdots\!25}a^{12}-\frac{62\!\cdots\!71}{25\!\cdots\!75}a^{11}-\frac{88\!\cdots\!86}{43\!\cdots\!75}a^{10}-\frac{63\!\cdots\!92}{48\!\cdots\!25}a^{9}+\frac{11\!\cdots\!78}{43\!\cdots\!75}a^{8}+\frac{73\!\cdots\!57}{36\!\cdots\!25}a^{7}-\frac{89\!\cdots\!71}{43\!\cdots\!75}a^{6}-\frac{10\!\cdots\!38}{39\!\cdots\!25}a^{5}+\frac{76\!\cdots\!99}{43\!\cdots\!75}a^{4}+\frac{11\!\cdots\!52}{33\!\cdots\!75}a^{3}+\frac{28\!\cdots\!49}{43\!\cdots\!75}a^{2}-\frac{25\!\cdots\!89}{56\!\cdots\!75}a-\frac{31\!\cdots\!33}{43\!\cdots\!75}$, $\frac{16\!\cdots\!68}{33\!\cdots\!75}a^{24}-\frac{66\!\cdots\!24}{87\!\cdots\!35}a^{23}-\frac{10\!\cdots\!07}{12\!\cdots\!05}a^{22}+\frac{26\!\cdots\!33}{43\!\cdots\!75}a^{21}+\frac{23\!\cdots\!23}{87\!\cdots\!35}a^{20}+\frac{53\!\cdots\!91}{43\!\cdots\!75}a^{19}-\frac{29\!\cdots\!89}{25\!\cdots\!75}a^{18}-\frac{29\!\cdots\!86}{43\!\cdots\!75}a^{17}+\frac{51\!\cdots\!23}{85\!\cdots\!75}a^{16}-\frac{28\!\cdots\!19}{62\!\cdots\!25}a^{15}-\frac{62\!\cdots\!07}{43\!\cdots\!75}a^{14}-\frac{15\!\cdots\!84}{43\!\cdots\!75}a^{13}+\frac{22\!\cdots\!38}{62\!\cdots\!25}a^{12}+\frac{48\!\cdots\!77}{25\!\cdots\!75}a^{11}-\frac{29\!\cdots\!13}{43\!\cdots\!75}a^{10}-\frac{30\!\cdots\!41}{62\!\cdots\!25}a^{9}+\frac{42\!\cdots\!51}{60\!\cdots\!75}a^{8}+\frac{28\!\cdots\!57}{36\!\cdots\!25}a^{7}-\frac{50\!\cdots\!59}{87\!\cdots\!35}a^{6}-\frac{39\!\cdots\!38}{43\!\cdots\!75}a^{5}+\frac{17\!\cdots\!16}{43\!\cdots\!75}a^{4}+\frac{49\!\cdots\!57}{43\!\cdots\!75}a^{3}+\frac{15\!\cdots\!24}{43\!\cdots\!75}a^{2}-\frac{12\!\cdots\!13}{12\!\cdots\!05}a+\frac{34\!\cdots\!08}{43\!\cdots\!75}$, $\frac{97\!\cdots\!26}{43\!\cdots\!75}a^{24}-\frac{20\!\cdots\!61}{39\!\cdots\!25}a^{23}-\frac{79\!\cdots\!02}{62\!\cdots\!25}a^{22}+\frac{13\!\cdots\!51}{43\!\cdots\!75}a^{21}+\frac{44\!\cdots\!84}{43\!\cdots\!75}a^{20}-\frac{49\!\cdots\!75}{13\!\cdots\!39}a^{19}-\frac{23\!\cdots\!11}{25\!\cdots\!75}a^{18}+\frac{60\!\cdots\!93}{43\!\cdots\!75}a^{17}+\frac{50\!\cdots\!94}{17\!\cdots\!15}a^{16}-\frac{23\!\cdots\!86}{56\!\cdots\!75}a^{15}-\frac{19\!\cdots\!91}{39\!\cdots\!25}a^{14}+\frac{29\!\cdots\!76}{87\!\cdots\!35}a^{13}+\frac{10\!\cdots\!22}{62\!\cdots\!25}a^{12}-\frac{22\!\cdots\!03}{19\!\cdots\!75}a^{11}-\frac{13\!\cdots\!24}{43\!\cdots\!75}a^{10}+\frac{11\!\cdots\!33}{62\!\cdots\!25}a^{9}+\frac{21\!\cdots\!36}{43\!\cdots\!75}a^{8}+\frac{35\!\cdots\!24}{36\!\cdots\!25}a^{7}-\frac{30\!\cdots\!67}{60\!\cdots\!75}a^{6}-\frac{86\!\cdots\!76}{43\!\cdots\!75}a^{5}+\frac{42\!\cdots\!92}{87\!\cdots\!35}a^{4}+\frac{15\!\cdots\!08}{43\!\cdots\!75}a^{3}-\frac{95\!\cdots\!57}{43\!\cdots\!75}a^{2}-\frac{93\!\cdots\!24}{62\!\cdots\!25}a+\frac{31\!\cdots\!43}{87\!\cdots\!35}$, $\frac{11\!\cdots\!54}{87\!\cdots\!35}a^{24}-\frac{18\!\cdots\!68}{43\!\cdots\!75}a^{23}+\frac{20\!\cdots\!69}{62\!\cdots\!25}a^{22}+\frac{64\!\cdots\!62}{43\!\cdots\!75}a^{21}+\frac{19\!\cdots\!87}{43\!\cdots\!75}a^{20}-\frac{27\!\cdots\!62}{43\!\cdots\!75}a^{19}+\frac{47\!\cdots\!83}{51\!\cdots\!55}a^{18}-\frac{12\!\cdots\!04}{43\!\cdots\!75}a^{17}+\frac{14\!\cdots\!49}{85\!\cdots\!75}a^{16}-\frac{49\!\cdots\!98}{12\!\cdots\!05}a^{15}+\frac{40\!\cdots\!16}{43\!\cdots\!75}a^{14}+\frac{44\!\cdots\!83}{43\!\cdots\!75}a^{13}+\frac{10\!\cdots\!68}{12\!\cdots\!05}a^{12}-\frac{30\!\cdots\!51}{20\!\cdots\!75}a^{11}-\frac{18\!\cdots\!41}{43\!\cdots\!75}a^{10}+\frac{31\!\cdots\!81}{62\!\cdots\!25}a^{9}+\frac{94\!\cdots\!42}{43\!\cdots\!75}a^{8}-\frac{51\!\cdots\!62}{36\!\cdots\!25}a^{7}-\frac{55\!\cdots\!41}{33\!\cdots\!75}a^{6}+\frac{32\!\cdots\!11}{60\!\cdots\!75}a^{5}+\frac{92\!\cdots\!53}{43\!\cdots\!75}a^{4}+\frac{42\!\cdots\!96}{87\!\cdots\!35}a^{3}-\frac{53\!\cdots\!34}{43\!\cdots\!75}a^{2}+\frac{23\!\cdots\!83}{62\!\cdots\!25}a-\frac{16\!\cdots\!61}{43\!\cdots\!75}$, $\frac{36\!\cdots\!77}{43\!\cdots\!75}a^{24}-\frac{74\!\cdots\!18}{43\!\cdots\!75}a^{23}-\frac{62\!\cdots\!26}{62\!\cdots\!25}a^{22}+\frac{50\!\cdots\!11}{43\!\cdots\!75}a^{21}+\frac{17\!\cdots\!57}{43\!\cdots\!75}a^{20}-\frac{13\!\cdots\!04}{43\!\cdots\!75}a^{19}-\frac{87\!\cdots\!92}{25\!\cdots\!75}a^{18}+\frac{31\!\cdots\!43}{43\!\cdots\!75}a^{17}+\frac{90\!\cdots\!63}{85\!\cdots\!75}a^{16}-\frac{59\!\cdots\!59}{48\!\cdots\!25}a^{15}-\frac{19\!\cdots\!18}{87\!\cdots\!35}a^{14}+\frac{46\!\cdots\!61}{43\!\cdots\!75}a^{13}+\frac{38\!\cdots\!44}{62\!\cdots\!25}a^{12}-\frac{27\!\cdots\!22}{10\!\cdots\!71}a^{11}-\frac{11\!\cdots\!56}{87\!\cdots\!35}a^{10}-\frac{55\!\cdots\!52}{62\!\cdots\!25}a^{9}+\frac{70\!\cdots\!56}{43\!\cdots\!75}a^{8}+\frac{24\!\cdots\!39}{36\!\cdots\!25}a^{7}-\frac{80\!\cdots\!93}{43\!\cdots\!75}a^{6}-\frac{40\!\cdots\!61}{43\!\cdots\!75}a^{5}+\frac{52\!\cdots\!82}{33\!\cdots\!75}a^{4}+\frac{69\!\cdots\!21}{43\!\cdots\!75}a^{3}-\frac{22\!\cdots\!32}{43\!\cdots\!75}a^{2}-\frac{37\!\cdots\!87}{62\!\cdots\!25}a+\frac{10\!\cdots\!98}{43\!\cdots\!75}$, $\frac{80\!\cdots\!91}{43\!\cdots\!75}a^{24}-\frac{14\!\cdots\!02}{43\!\cdots\!75}a^{23}-\frac{19\!\cdots\!09}{62\!\cdots\!25}a^{22}+\frac{21\!\cdots\!82}{87\!\cdots\!35}a^{21}+\frac{43\!\cdots\!93}{43\!\cdots\!75}a^{20}+\frac{11\!\cdots\!91}{43\!\cdots\!75}a^{19}-\frac{16\!\cdots\!01}{25\!\cdots\!75}a^{18}-\frac{10\!\cdots\!31}{87\!\cdots\!35}a^{17}+\frac{19\!\cdots\!08}{85\!\cdots\!75}a^{16}-\frac{13\!\cdots\!01}{62\!\cdots\!25}a^{15}-\frac{23\!\cdots\!69}{43\!\cdots\!75}a^{14}+\frac{85\!\cdots\!76}{43\!\cdots\!75}a^{13}+\frac{86\!\cdots\!82}{62\!\cdots\!25}a^{12}-\frac{39\!\cdots\!46}{25\!\cdots\!75}a^{11}-\frac{12\!\cdots\!06}{43\!\cdots\!75}a^{10}-\frac{15\!\cdots\!21}{12\!\cdots\!05}a^{9}+\frac{27\!\cdots\!52}{87\!\cdots\!35}a^{8}+\frac{18\!\cdots\!12}{73\!\cdots\!65}a^{7}-\frac{13\!\cdots\!82}{43\!\cdots\!75}a^{6}-\frac{26\!\cdots\!89}{87\!\cdots\!35}a^{5}+\frac{80\!\cdots\!87}{33\!\cdots\!75}a^{4}+\frac{18\!\cdots\!23}{43\!\cdots\!75}a^{3}+\frac{33\!\cdots\!62}{87\!\cdots\!35}a^{2}-\frac{50\!\cdots\!93}{62\!\cdots\!25}a+\frac{76\!\cdots\!58}{43\!\cdots\!75}$, $\frac{36\!\cdots\!89}{48\!\cdots\!25}a^{24}-\frac{87\!\cdots\!67}{62\!\cdots\!25}a^{23}-\frac{12\!\cdots\!92}{12\!\cdots\!25}a^{22}+\frac{60\!\cdots\!07}{62\!\cdots\!25}a^{21}+\frac{24\!\cdots\!73}{62\!\cdots\!25}a^{20}+\frac{18\!\cdots\!46}{25\!\cdots\!01}a^{19}-\frac{85\!\cdots\!67}{36\!\cdots\!25}a^{18}-\frac{52\!\cdots\!94}{62\!\cdots\!25}a^{17}+\frac{23\!\cdots\!58}{24\!\cdots\!45}a^{16}-\frac{85\!\cdots\!97}{89\!\cdots\!75}a^{15}-\frac{12\!\cdots\!77}{62\!\cdots\!25}a^{14}-\frac{17\!\cdots\!49}{12\!\cdots\!05}a^{13}+\frac{51\!\cdots\!34}{89\!\cdots\!75}a^{12}-\frac{40\!\cdots\!43}{36\!\cdots\!25}a^{11}-\frac{65\!\cdots\!08}{62\!\cdots\!25}a^{10}-\frac{44\!\cdots\!34}{89\!\cdots\!75}a^{9}+\frac{82\!\cdots\!57}{62\!\cdots\!25}a^{8}+\frac{72\!\cdots\!54}{75\!\cdots\!25}a^{7}-\frac{74\!\cdots\!32}{62\!\cdots\!25}a^{6}-\frac{76\!\cdots\!17}{62\!\cdots\!25}a^{5}+\frac{11\!\cdots\!92}{12\!\cdots\!05}a^{4}+\frac{10\!\cdots\!71}{62\!\cdots\!25}a^{3}+\frac{88\!\cdots\!71}{62\!\cdots\!25}a^{2}-\frac{34\!\cdots\!18}{89\!\cdots\!75}a-\frac{39\!\cdots\!18}{12\!\cdots\!05}$, $\frac{52\!\cdots\!92}{11\!\cdots\!75}a^{24}-\frac{49\!\cdots\!14}{11\!\cdots\!75}a^{23}-\frac{19\!\cdots\!68}{13\!\cdots\!25}a^{22}+\frac{12\!\cdots\!34}{23\!\cdots\!55}a^{21}+\frac{33\!\cdots\!31}{11\!\cdots\!75}a^{20}+\frac{26\!\cdots\!42}{10\!\cdots\!25}a^{19}-\frac{58\!\cdots\!99}{53\!\cdots\!75}a^{18}-\frac{23\!\cdots\!93}{18\!\cdots\!35}a^{17}+\frac{45\!\cdots\!12}{85\!\cdots\!75}a^{16}-\frac{12\!\cdots\!57}{16\!\cdots\!25}a^{15}-\frac{20\!\cdots\!53}{11\!\cdots\!75}a^{14}-\frac{10\!\cdots\!03}{11\!\cdots\!75}a^{13}+\frac{56\!\cdots\!84}{16\!\cdots\!25}a^{12}+\frac{15\!\cdots\!93}{69\!\cdots\!75}a^{11}-\frac{83\!\cdots\!87}{11\!\cdots\!75}a^{10}-\frac{26\!\cdots\!03}{33\!\cdots\!65}a^{9}+\frac{13\!\cdots\!44}{23\!\cdots\!55}a^{8}+\frac{17\!\cdots\!51}{15\!\cdots\!65}a^{7}-\frac{36\!\cdots\!14}{11\!\cdots\!75}a^{6}-\frac{30\!\cdots\!88}{23\!\cdots\!55}a^{5}+\frac{11\!\cdots\!42}{11\!\cdots\!75}a^{4}+\frac{16\!\cdots\!16}{11\!\cdots\!75}a^{3}+\frac{14\!\cdots\!13}{19\!\cdots\!55}a^{2}-\frac{39\!\cdots\!76}{16\!\cdots\!25}a-\frac{61\!\cdots\!89}{11\!\cdots\!75}$, $\frac{16\!\cdots\!23}{43\!\cdots\!75}a^{24}-\frac{29\!\cdots\!97}{43\!\cdots\!75}a^{23}-\frac{34\!\cdots\!84}{62\!\cdots\!25}a^{22}+\frac{20\!\cdots\!34}{43\!\cdots\!75}a^{21}+\frac{67\!\cdots\!66}{33\!\cdots\!75}a^{20}+\frac{25\!\cdots\!69}{43\!\cdots\!75}a^{19}-\frac{26\!\cdots\!03}{25\!\cdots\!75}a^{18}-\frac{19\!\cdots\!73}{39\!\cdots\!25}a^{17}+\frac{40\!\cdots\!07}{85\!\cdots\!75}a^{16}-\frac{26\!\cdots\!88}{62\!\cdots\!25}a^{15}-\frac{89\!\cdots\!19}{87\!\cdots\!35}a^{14}-\frac{43\!\cdots\!97}{33\!\cdots\!75}a^{13}+\frac{17\!\cdots\!01}{62\!\cdots\!25}a^{12}-\frac{13\!\cdots\!32}{51\!\cdots\!55}a^{11}-\frac{91\!\cdots\!50}{17\!\cdots\!07}a^{10}-\frac{14\!\cdots\!71}{48\!\cdots\!25}a^{9}+\frac{26\!\cdots\!94}{43\!\cdots\!75}a^{8}+\frac{19\!\cdots\!86}{36\!\cdots\!25}a^{7}-\frac{23\!\cdots\!67}{43\!\cdots\!75}a^{6}-\frac{25\!\cdots\!44}{39\!\cdots\!25}a^{5}+\frac{18\!\cdots\!69}{43\!\cdots\!75}a^{4}+\frac{37\!\cdots\!79}{43\!\cdots\!75}a^{3}+\frac{63\!\cdots\!67}{43\!\cdots\!75}a^{2}-\frac{84\!\cdots\!98}{56\!\cdots\!75}a-\frac{47\!\cdots\!23}{43\!\cdots\!75}$, $\frac{59\!\cdots\!93}{43\!\cdots\!75}a^{24}-\frac{12\!\cdots\!14}{43\!\cdots\!75}a^{23}-\frac{42\!\cdots\!38}{62\!\cdots\!25}a^{22}+\frac{75\!\cdots\!52}{43\!\cdots\!75}a^{21}+\frac{25\!\cdots\!76}{39\!\cdots\!25}a^{20}-\frac{24\!\cdots\!39}{43\!\cdots\!75}a^{19}-\frac{79\!\cdots\!13}{25\!\cdots\!75}a^{18}-\frac{57\!\cdots\!14}{43\!\cdots\!75}a^{17}+\frac{13\!\cdots\!03}{77\!\cdots\!25}a^{16}-\frac{14\!\cdots\!33}{62\!\cdots\!25}a^{15}-\frac{11\!\cdots\!71}{43\!\cdots\!75}a^{14}+\frac{23\!\cdots\!46}{43\!\cdots\!75}a^{13}+\frac{61\!\cdots\!16}{62\!\cdots\!25}a^{12}-\frac{14\!\cdots\!74}{25\!\cdots\!75}a^{11}-\frac{70\!\cdots\!99}{43\!\cdots\!75}a^{10}-\frac{24\!\cdots\!19}{56\!\cdots\!75}a^{9}+\frac{10\!\cdots\!72}{43\!\cdots\!75}a^{8}+\frac{31\!\cdots\!98}{36\!\cdots\!25}a^{7}-\frac{95\!\cdots\!54}{43\!\cdots\!75}a^{6}-\frac{59\!\cdots\!02}{43\!\cdots\!75}a^{5}+\frac{73\!\cdots\!06}{36\!\cdots\!75}a^{4}+\frac{98\!\cdots\!74}{43\!\cdots\!75}a^{3}-\frac{93\!\cdots\!74}{43\!\cdots\!75}a^{2}-\frac{21\!\cdots\!92}{48\!\cdots\!25}a+\frac{29\!\cdots\!73}{43\!\cdots\!75}$, $\frac{50\!\cdots\!48}{36\!\cdots\!75}a^{24}-\frac{19\!\cdots\!62}{43\!\cdots\!75}a^{23}+\frac{20\!\cdots\!41}{62\!\cdots\!25}a^{22}+\frac{71\!\cdots\!34}{43\!\cdots\!75}a^{21}+\frac{21\!\cdots\!18}{43\!\cdots\!75}a^{20}-\frac{28\!\cdots\!46}{43\!\cdots\!75}a^{19}+\frac{14\!\cdots\!67}{25\!\cdots\!75}a^{18}+\frac{23\!\cdots\!82}{43\!\cdots\!75}a^{17}+\frac{11\!\cdots\!94}{65\!\cdots\!75}a^{16}-\frac{26\!\cdots\!43}{62\!\cdots\!25}a^{15}+\frac{60\!\cdots\!52}{87\!\cdots\!35}a^{14}+\frac{61\!\cdots\!24}{43\!\cdots\!75}a^{13}+\frac{56\!\cdots\!46}{62\!\cdots\!25}a^{12}-\frac{77\!\cdots\!91}{51\!\cdots\!55}a^{11}-\frac{42\!\cdots\!99}{79\!\cdots\!85}a^{10}+\frac{39\!\cdots\!12}{62\!\cdots\!25}a^{9}+\frac{97\!\cdots\!14}{43\!\cdots\!75}a^{8}-\frac{53\!\cdots\!49}{36\!\cdots\!25}a^{7}-\frac{78\!\cdots\!27}{43\!\cdots\!75}a^{6}+\frac{29\!\cdots\!01}{43\!\cdots\!75}a^{5}+\frac{10\!\cdots\!89}{43\!\cdots\!75}a^{4}+\frac{17\!\cdots\!09}{43\!\cdots\!75}a^{3}-\frac{57\!\cdots\!33}{43\!\cdots\!75}a^{2}+\frac{30\!\cdots\!77}{62\!\cdots\!25}a-\frac{25\!\cdots\!18}{43\!\cdots\!75}$, $\frac{61\!\cdots\!76}{43\!\cdots\!75}a^{24}-\frac{11\!\cdots\!72}{33\!\cdots\!75}a^{23}+\frac{56\!\cdots\!03}{62\!\cdots\!25}a^{22}+\frac{66\!\cdots\!06}{43\!\cdots\!75}a^{21}+\frac{28\!\cdots\!64}{43\!\cdots\!75}a^{20}-\frac{30\!\cdots\!16}{87\!\cdots\!35}a^{19}+\frac{35\!\cdots\!14}{25\!\cdots\!75}a^{18}-\frac{11\!\cdots\!72}{43\!\cdots\!75}a^{17}+\frac{29\!\cdots\!31}{17\!\cdots\!15}a^{16}-\frac{16\!\cdots\!61}{62\!\cdots\!25}a^{15}-\frac{42\!\cdots\!61}{43\!\cdots\!75}a^{14}-\frac{13\!\cdots\!73}{87\!\cdots\!35}a^{13}+\frac{59\!\cdots\!82}{62\!\cdots\!25}a^{12}-\frac{23\!\cdots\!13}{35\!\cdots\!75}a^{11}-\frac{36\!\cdots\!39}{43\!\cdots\!75}a^{10}-\frac{62\!\cdots\!02}{62\!\cdots\!25}a^{9}+\frac{75\!\cdots\!36}{43\!\cdots\!75}a^{8}+\frac{18\!\cdots\!04}{36\!\cdots\!25}a^{7}-\frac{28\!\cdots\!96}{43\!\cdots\!75}a^{6}-\frac{78\!\cdots\!12}{60\!\cdots\!75}a^{5}+\frac{15\!\cdots\!30}{17\!\cdots\!07}a^{4}+\frac{66\!\cdots\!78}{43\!\cdots\!75}a^{3}+\frac{43\!\cdots\!73}{43\!\cdots\!75}a^{2}+\frac{28\!\cdots\!91}{62\!\cdots\!25}a-\frac{28\!\cdots\!68}{17\!\cdots\!07}$
| sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
| |
| Regulator: | \( 222319295.2100307 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{12}\cdot 222319295.2100307 \cdot 4}{2\cdot\sqrt{42581619494519898305269398418425099361}}\cr\approx \mathstrut & 0.515922515220275 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^25 - 2*x^24 - x^23 + 13*x^22 + 50*x^21 + 3*x^20 - 29*x^19 - 5*x^18 + 128*x^17 - 146*x^16 - 239*x^15 + 32*x^14 + 747*x^13 - 262*x^12 - 1333*x^11 - 456*x^10 + 1762*x^9 + 962*x^8 - 1685*x^7 - 1308*x^6 + 1439*x^5 + 1916*x^4 - 95*x^3 - 410*x^2 + 94*x - 1)
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
# self-contained Pari/GP code snippet to compute the analytic class number formula
K = bnfinit(x^25 - 2*x^24 - x^23 + 13*x^22 + 50*x^21 + 3*x^20 - 29*x^19 - 5*x^18 + 128*x^17 - 146*x^16 - 239*x^15 + 32*x^14 + 747*x^13 - 262*x^12 - 1333*x^11 - 456*x^10 + 1762*x^9 + 962*x^8 - 1685*x^7 - 1308*x^6 + 1439*x^5 + 1916*x^4 - 95*x^3 - 410*x^2 + 94*x - 1, 1);
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
/* self-contained Magma code snippet to compute the analytic class number formula */
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^25 - 2*x^24 - x^23 + 13*x^22 + 50*x^21 + 3*x^20 - 29*x^19 - 5*x^18 + 128*x^17 - 146*x^16 - 239*x^15 + 32*x^14 + 747*x^13 - 262*x^12 - 1333*x^11 - 456*x^10 + 1762*x^9 + 962*x^8 - 1685*x^7 - 1308*x^6 + 1439*x^5 + 1916*x^4 - 95*x^3 - 410*x^2 + 94*x - 1);
OK := Integers(K); DK := Discriminant(OK);
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
hK := #clK; wK := #TorsionSubgroup(UK);
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
# self-contained Oscar code snippet to compute the analytic class number formula
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^25 - 2*x^24 - x^23 + 13*x^22 + 50*x^21 + 3*x^20 - 29*x^19 - 5*x^18 + 128*x^17 - 146*x^16 - 239*x^15 + 32*x^14 + 747*x^13 - 262*x^12 - 1333*x^11 - 456*x^10 + 1762*x^9 + 962*x^8 - 1685*x^7 - 1308*x^6 + 1439*x^5 + 1916*x^4 - 95*x^3 - 410*x^2 + 94*x - 1);
OK = ring_of_integers(K); DK = discriminant(OK);
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
hK = order(clK); wK = torsion_units_order(K);
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
Galois group
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
magma: G = GaloisGroup(K);
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
| A solvable group of order 50 |
| The 14 conjugacy class representatives for $D_{25}$ |
| Character table for $D_{25}$ |
Intermediate fields
| 5.1.1868689.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
sage: K.subfields()[1:-1]
gp: L = nfsubfields(K); L[2..length(b)]
magma: L := Subfields(K); L[2..#L];
oscar: subfields(K)[2:end-1]
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | $25$ | $25$ | ${\href{/padicField/5.2.0.1}{2} }^{12}{,}\,{\href{/padicField/5.1.0.1}{1} }$ | ${\href{/padicField/7.2.0.1}{2} }^{12}{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.2.0.1}{2} }^{12}{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.2.0.1}{2} }^{12}{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.2.0.1}{2} }^{12}{,}\,{\href{/padicField/17.1.0.1}{1} }$ | $25$ | ${\href{/padicField/23.5.0.1}{5} }^{5}$ | $25$ | ${\href{/padicField/31.5.0.1}{5} }^{5}$ | ${\href{/padicField/37.2.0.1}{2} }^{12}{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.2.0.1}{2} }^{12}{,}\,{\href{/padicField/41.1.0.1}{1} }$ | $25$ | ${\href{/padicField/47.2.0.1}{2} }^{12}{,}\,{\href{/padicField/47.1.0.1}{1} }$ | $25$ | $25$ |
Cycle lengths which are repeated in a cycle type are indicated by exponents.
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(1367\)
| $\Q_{1367}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
Artin representations
Data is given for all irreducible
representations of the Galois group for the Galois closure
of this field. Those marked with * are summands in the
permutation representation coming from this field. Representations
which appear with multiplicity greater than one are indicated
by exponents on the *.