Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - x^14 - 138*x^13 - 80*x^12 + 6278*x^11 + 13450*x^10 - 98056*x^9 - 360148*x^8 + 54921*x^7 + 1300271*x^6 + 477210*x^5 - 1783924*x^4 - 627480*x^3 + 972256*x^2 + 154496*x - 141824)
gp: K = bnfinit(y^15 - y^14 - 138*y^13 - 80*y^12 + 6278*y^11 + 13450*y^10 - 98056*y^9 - 360148*y^8 + 54921*y^7 + 1300271*y^6 + 477210*y^5 - 1783924*y^4 - 627480*y^3 + 972256*y^2 + 154496*y - 141824, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^15 - x^14 - 138*x^13 - 80*x^12 + 6278*x^11 + 13450*x^10 - 98056*x^9 - 360148*x^8 + 54921*x^7 + 1300271*x^6 + 477210*x^5 - 1783924*x^4 - 627480*x^3 + 972256*x^2 + 154496*x - 141824);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^15 - x^14 - 138*x^13 - 80*x^12 + 6278*x^11 + 13450*x^10 - 98056*x^9 - 360148*x^8 + 54921*x^7 + 1300271*x^6 + 477210*x^5 - 1783924*x^4 - 627480*x^3 + 972256*x^2 + 154496*x - 141824)
\( x^{15} - x^{14} - 138 x^{13} - 80 x^{12} + 6278 x^{11} + 13450 x^{10} - 98056 x^{9} - 360148 x^{8} + \cdots - 141824 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $15$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[15, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(2375614325883809574306005975647441\)
\(\medspace = 11^{12}\cdot 31^{14}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(167.90\) | 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: | $11^{4/5}31^{14/15}\approx 167.90039932398548$ | ||
| Ramified primes: |
\(11\), \(31\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Gal(K/\Q) }$: | $15$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(341=11\cdot 31\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{341}(1,·)$, $\chi_{341}(67,·)$, $\chi_{341}(324,·)$, $\chi_{341}(71,·)$, $\chi_{341}(202,·)$, $\chi_{341}(267,·)$, $\chi_{341}(289,·)$, $\chi_{341}(157,·)$, $\chi_{341}(20,·)$, $\chi_{341}(97,·)$, $\chi_{341}(235,·)$, $\chi_{341}(56,·)$, $\chi_{341}(225,·)$, $\chi_{341}(59,·)$, $\chi_{341}(317,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{8}a^{4}-\frac{1}{4}a^{3}-\frac{1}{8}a^{2}+\frac{1}{4}a$, $\frac{1}{8}a^{5}-\frac{1}{8}a^{3}$, $\frac{1}{16}a^{6}-\frac{1}{16}a^{5}-\frac{1}{16}a^{4}-\frac{3}{16}a^{3}+\frac{1}{4}a$, $\frac{1}{32}a^{7}-\frac{1}{16}a^{5}-\frac{3}{32}a^{3}-\frac{3}{8}a-\frac{1}{2}$, $\frac{1}{128}a^{8}+\frac{1}{64}a^{6}+\frac{1}{128}a^{4}-\frac{1}{4}a^{3}-\frac{1}{32}a^{2}-\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{256}a^{9}-\frac{1}{256}a^{8}+\frac{1}{128}a^{7}-\frac{1}{128}a^{6}+\frac{1}{256}a^{5}-\frac{1}{256}a^{4}+\frac{7}{64}a^{3}-\frac{15}{64}a^{2}-\frac{1}{8}a+\frac{1}{4}$, $\frac{1}{512}a^{10}-\frac{1}{512}a^{9}-\frac{1}{256}a^{7}+\frac{13}{512}a^{6}-\frac{17}{512}a^{5}+\frac{5}{256}a^{4}-\frac{11}{128}a^{3}-\frac{3}{64}a^{2}-\frac{1}{8}a-\frac{1}{4}$, $\frac{1}{1024}a^{11}-\frac{1}{1024}a^{9}-\frac{1}{512}a^{8}+\frac{11}{1024}a^{7}-\frac{1}{256}a^{6}+\frac{57}{1024}a^{5}+\frac{15}{512}a^{4}+\frac{63}{256}a^{3}-\frac{19}{128}a^{2}-\frac{1}{16}a+\frac{3}{8}$, $\frac{1}{4096}a^{12}-\frac{1}{2048}a^{11}+\frac{1}{4096}a^{10}-\frac{1}{2048}a^{9}-\frac{9}{4096}a^{8}-\frac{15}{2048}a^{7}+\frac{107}{4096}a^{6}+\frac{37}{2048}a^{5}+\frac{63}{1024}a^{4}+\frac{107}{512}a^{3}-\frac{27}{128}a^{2}-\frac{15}{32}a-\frac{3}{8}$, $\frac{1}{16384}a^{13}-\frac{1}{16384}a^{12}+\frac{7}{16384}a^{11}+\frac{15}{16384}a^{10}-\frac{19}{16384}a^{9}-\frac{39}{16384}a^{8}+\frac{37}{16384}a^{7}+\frac{133}{16384}a^{6}+\frac{263}{8192}a^{5}-\frac{67}{4096}a^{4}+\frac{43}{2048}a^{3}+\frac{85}{512}a^{2}-\frac{47}{128}a-\frac{15}{32}$, $\frac{1}{56\!\cdots\!84}a^{14}-\frac{19765035243}{28\!\cdots\!92}a^{13}-\frac{91428392447}{14\!\cdots\!96}a^{12}+\frac{459242087223}{14\!\cdots\!96}a^{11}-\frac{2621394266115}{28\!\cdots\!92}a^{10}-\frac{1019217603839}{704727327899648}a^{9}+\frac{521378543325}{352363663949824}a^{8}-\frac{6924064513625}{14\!\cdots\!96}a^{7}+\frac{125852046007869}{56\!\cdots\!84}a^{6}+\frac{80292896350071}{28\!\cdots\!92}a^{5}-\frac{75586316127675}{14\!\cdots\!96}a^{4}-\frac{19830503956483}{704727327899648}a^{3}-\frac{20169071449817}{176181831974912}a^{2}+\frac{6886292425167}{44045457993728}a-\frac{3380923733569}{11011364498432}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
Class group and class number
$C_{5}$, which has order $5$ (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: | $14$ | 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: |
$\frac{42139199063}{352363663949824}a^{14}+\frac{36084644759}{176181831974912}a^{13}-\frac{1465575413551}{88090915987456}a^{12}-\frac{4737561332177}{88090915987456}a^{11}+\frac{122835542212463}{176181831974912}a^{10}+\frac{38886509657369}{11011364498432}a^{9}-\frac{271466117531919}{44045457993728}a^{8}-\frac{59\!\cdots\!41}{88090915987456}a^{7}-\frac{42\!\cdots\!05}{352363663949824}a^{6}+\frac{79\!\cdots\!17}{176181831974912}a^{5}+\frac{24\!\cdots\!55}{88090915987456}a^{4}+\frac{43\!\cdots\!87}{44045457993728}a^{3}-\frac{16\!\cdots\!47}{11011364498432}a^{2}-\frac{131210140758031}{2752841124608}a+\frac{7963436788177}{688210281152}$, $\frac{478100471297}{28\!\cdots\!92}a^{14}+\frac{54623103989}{14\!\cdots\!96}a^{13}-\frac{16430299467247}{704727327899648}a^{12}-\frac{29822344282553}{704727327899648}a^{11}+\frac{14\!\cdots\!93}{14\!\cdots\!96}a^{10}+\frac{12\!\cdots\!53}{352363663949824}a^{9}-\frac{21\!\cdots\!71}{176181831974912}a^{8}-\frac{53\!\cdots\!05}{704727327899648}a^{7}-\frac{24\!\cdots\!99}{28\!\cdots\!92}a^{6}+\frac{14\!\cdots\!63}{14\!\cdots\!96}a^{5}+\frac{15\!\cdots\!13}{704727327899648}a^{4}+\frac{508472259470141}{352363663949824}a^{3}-\frac{92\!\cdots\!85}{88090915987456}a^{2}-\frac{97282609645681}{22022728996864}a+\frac{68055304846719}{5505682249216}$, $\frac{66573577849}{352363663949824}a^{14}-\frac{53601248843}{176181831974912}a^{13}-\frac{2251086389063}{88090915987456}a^{12}+\frac{19312454099}{88090915987456}a^{11}+\frac{200818551306069}{176181831974912}a^{10}+\frac{79789617303103}{44045457993728}a^{9}-\frac{386309403479483}{22022728996864}a^{8}-\frac{46\!\cdots\!25}{88090915987456}a^{7}+\frac{38\!\cdots\!25}{352363663949824}a^{6}+\frac{21\!\cdots\!59}{176181831974912}a^{5}+\frac{754675935134301}{88090915987456}a^{4}-\frac{21\!\cdots\!87}{44045457993728}a^{3}-\frac{10570798507825}{11011364498432}a^{2}-\frac{87716936611977}{2752841124608}a+\frac{13622196091815}{688210281152}$, $\frac{168345591929}{28\!\cdots\!92}a^{14}-\frac{241639685379}{14\!\cdots\!96}a^{13}-\frac{5398759179087}{704727327899648}a^{12}+\frac{6267559199287}{704727327899648}a^{11}+\frac{457180085315589}{14\!\cdots\!96}a^{10}+\frac{79364026970413}{352363663949824}a^{9}-\frac{847423604175657}{176181831974912}a^{8}-\frac{78\!\cdots\!69}{704727327899648}a^{7}+\frac{46\!\cdots\!57}{28\!\cdots\!92}a^{6}+\frac{26\!\cdots\!95}{14\!\cdots\!96}a^{5}+\frac{96\!\cdots\!09}{704727327899648}a^{4}+\frac{526357929515509}{352363663949824}a^{3}-\frac{15\!\cdots\!73}{88090915987456}a^{2}-\frac{147910914816265}{22022728996864}a+\frac{29102557066727}{5505682249216}$, $\frac{417896273907}{28\!\cdots\!92}a^{14}-\frac{440446215825}{14\!\cdots\!96}a^{13}-\frac{14541604595789}{704727327899648}a^{12}+\frac{7639302069493}{704727327899648}a^{11}+\frac{13\!\cdots\!75}{14\!\cdots\!96}a^{10}+\frac{364198815911571}{352363663949824}a^{9}-\frac{33\!\cdots\!61}{176181831974912}a^{8}-\frac{30\!\cdots\!87}{704727327899648}a^{7}+\frac{25\!\cdots\!79}{28\!\cdots\!92}a^{6}+\frac{46\!\cdots\!65}{14\!\cdots\!96}a^{5}+\frac{11\!\cdots\!83}{704727327899648}a^{4}-\frac{19\!\cdots\!05}{352363663949824}a^{3}-\frac{19\!\cdots\!19}{88090915987456}a^{2}+\frac{49\!\cdots\!61}{22022728996864}a+\frac{502619953283533}{5505682249216}$, $\frac{67602859203}{14\!\cdots\!96}a^{14}+\frac{88008641655}{704727327899648}a^{13}-\frac{2491463224745}{352363663949824}a^{12}-\frac{9283921323799}{352363663949824}a^{11}+\frac{226580074197599}{704727327899648}a^{10}+\frac{285281657249977}{176181831974912}a^{9}-\frac{168090440382979}{44045457993728}a^{8}-\frac{11\!\cdots\!95}{352363663949824}a^{7}-\frac{49\!\cdots\!41}{14\!\cdots\!96}a^{6}+\frac{55\!\cdots\!81}{704727327899648}a^{5}+\frac{45\!\cdots\!23}{352363663949824}a^{4}-\frac{90\!\cdots\!13}{176181831974912}a^{3}-\frac{50\!\cdots\!31}{44045457993728}a^{2}-\frac{15630772997667}{11011364498432}a+\frac{46032982558381}{2752841124608}$, $\frac{2011021884997}{28\!\cdots\!92}a^{14}-\frac{164250881399}{14\!\cdots\!96}a^{13}-\frac{68709720741227}{704727327899648}a^{12}-\frac{99993039172253}{704727327899648}a^{11}+\frac{59\!\cdots\!89}{14\!\cdots\!96}a^{10}+\frac{46\!\cdots\!09}{352363663949824}a^{9}-\frac{93\!\cdots\!43}{176181831974912}a^{8}-\frac{21\!\cdots\!17}{704727327899648}a^{7}-\frac{83\!\cdots\!31}{28\!\cdots\!92}a^{6}+\frac{69\!\cdots\!75}{14\!\cdots\!96}a^{5}+\frac{60\!\cdots\!05}{704727327899648}a^{4}-\frac{46\!\cdots\!03}{352363663949824}a^{3}-\frac{46\!\cdots\!01}{88090915987456}a^{2}-\frac{384346295844213}{22022728996864}a+\frac{411096783675067}{5505682249216}$, $\frac{1311297282633}{28\!\cdots\!92}a^{14}-\frac{225755404275}{14\!\cdots\!96}a^{13}-\frac{45339113426719}{704727327899648}a^{12}-\frac{55785217765849}{704727327899648}a^{11}+\frac{40\!\cdots\!69}{14\!\cdots\!96}a^{10}+\frac{28\!\cdots\!29}{352363663949824}a^{9}-\frac{71\!\cdots\!77}{176181831974912}a^{8}-\frac{13\!\cdots\!05}{704727327899648}a^{7}-\frac{27\!\cdots\!83}{28\!\cdots\!92}a^{6}+\frac{75\!\cdots\!91}{14\!\cdots\!96}a^{5}+\frac{39\!\cdots\!25}{704727327899648}a^{4}-\frac{15\!\cdots\!51}{352363663949824}a^{3}-\frac{47\!\cdots\!29}{88090915987456}a^{2}+\frac{11\!\cdots\!99}{22022728996864}a+\frac{410590424985623}{5505682249216}$, $\frac{658395778735}{28\!\cdots\!92}a^{14}-\frac{60971000997}{14\!\cdots\!96}a^{13}-\frac{22559508271505}{704727327899648}a^{12}-\frac{31854635904487}{704727327899648}a^{11}+\frac{19\!\cdots\!19}{14\!\cdots\!96}a^{10}+\frac{15\!\cdots\!71}{352363663949824}a^{9}-\frac{31\!\cdots\!17}{176181831974912}a^{8}-\frac{67\!\cdots\!55}{704727327899648}a^{7}-\frac{24\!\cdots\!13}{28\!\cdots\!92}a^{6}+\frac{22\!\cdots\!53}{14\!\cdots\!96}a^{5}+\frac{17\!\cdots\!43}{704727327899648}a^{4}-\frac{16\!\cdots\!29}{352363663949824}a^{3}-\frac{12\!\cdots\!03}{88090915987456}a^{2}-\frac{60936263298687}{22022728996864}a+\frac{77394428881553}{5505682249216}$, $\frac{14319950093}{176181831974912}a^{14}+\frac{33711449029}{44045457993728}a^{13}-\frac{1149709772267}{88090915987456}a^{12}-\frac{10388308024771}{88090915987456}a^{11}+\frac{12414537271473}{22022728996864}a^{10}+\frac{521583516497713}{88090915987456}a^{9}-\frac{156914306824507}{88090915987456}a^{8}-\frac{89\!\cdots\!93}{88090915987456}a^{7}-\frac{35\!\cdots\!13}{176181831974912}a^{6}+\frac{98\!\cdots\!97}{88090915987456}a^{5}+\frac{21\!\cdots\!43}{44045457993728}a^{4}+\frac{12\!\cdots\!71}{22022728996864}a^{3}-\frac{17\!\cdots\!15}{5505682249216}a^{2}-\frac{45784860266211}{1376420562304}a+\frac{16237048050333}{344105140576}$, $\frac{20264842291}{176181831974912}a^{14}+\frac{31164859629}{44045457993728}a^{13}-\frac{1586777519549}{88090915987456}a^{12}-\frac{10182878337721}{88090915987456}a^{11}+\frac{17968990550677}{22022728996864}a^{10}+\frac{536039560772203}{88090915987456}a^{9}-\frac{608598437732125}{88090915987456}a^{8}-\frac{95\!\cdots\!51}{88090915987456}a^{7}-\frac{29\!\cdots\!79}{176181831974912}a^{6}+\frac{14\!\cdots\!03}{88090915987456}a^{5}+\frac{18\!\cdots\!09}{44045457993728}a^{4}-\frac{607110379822039}{22022728996864}a^{3}-\frac{14\!\cdots\!73}{5505682249216}a^{2}-\frac{11691687049629}{1376420562304}a+\frac{13929978367523}{344105140576}$, $\frac{176182098449}{704727327899648}a^{14}-\frac{96593605935}{352363663949824}a^{13}-\frac{5961879790709}{176181831974912}a^{12}-\frac{3289352822975}{176181831974912}a^{11}+\frac{524891741849345}{352363663949824}a^{10}+\frac{1145882101745}{344105140576}a^{9}-\frac{18\!\cdots\!39}{88090915987456}a^{8}-\frac{15\!\cdots\!99}{176181831974912}a^{7}-\frac{23\!\cdots\!83}{704727327899648}a^{6}+\frac{76\!\cdots\!03}{352363663949824}a^{5}+\frac{34\!\cdots\!89}{176181831974912}a^{4}-\frac{15\!\cdots\!99}{88090915987456}a^{3}-\frac{39\!\cdots\!33}{22022728996864}a^{2}+\frac{190613412408919}{5505682249216}a+\frac{48432227256519}{1376420562304}$, $\frac{390242678167}{704727327899648}a^{14}-\frac{23373844585}{352363663949824}a^{13}-\frac{13362147284323}{176181831974912}a^{12}-\frac{19867674342873}{176181831974912}a^{11}+\frac{11\!\cdots\!51}{352363663949824}a^{10}+\frac{28715216884327}{2752841124608}a^{9}-\frac{36\!\cdots\!73}{88090915987456}a^{8}-\frac{41\!\cdots\!69}{176181831974912}a^{7}-\frac{16\!\cdots\!53}{704727327899648}a^{6}+\frac{13\!\cdots\!57}{352363663949824}a^{5}+\frac{11\!\cdots\!27}{176181831974912}a^{4}-\frac{78\!\cdots\!41}{88090915987456}a^{3}-\frac{92\!\cdots\!71}{22022728996864}a^{2}-\frac{204852799382015}{5505682249216}a+\frac{97406407893777}{1376420562304}$, $\frac{1921333640815}{14\!\cdots\!96}a^{14}+\frac{137132635739}{704727327899648}a^{13}-\frac{66249842390577}{352363663949824}a^{12}-\frac{113704176947719}{352363663949824}a^{11}+\frac{57\!\cdots\!59}{704727327899648}a^{10}+\frac{48\!\cdots\!43}{176181831974912}a^{9}-\frac{90\!\cdots\!85}{88090915987456}a^{8}-\frac{21\!\cdots\!07}{352363663949824}a^{7}-\frac{85\!\cdots\!89}{14\!\cdots\!96}a^{6}+\frac{72\!\cdots\!89}{704727327899648}a^{5}+\frac{60\!\cdots\!51}{352363663949824}a^{4}-\frac{60\!\cdots\!37}{176181831974912}a^{3}-\frac{47\!\cdots\!51}{44045457993728}a^{2}-\frac{255783557290559}{11011364498432}a+\frac{419224752633489}{2752841124608}$
| 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: | \( 2170143402898142.5 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{15}\cdot(2\pi)^{0}\cdot 2170143402898142.5 \cdot 5}{2\cdot\sqrt{2375614325883809574306005975647441}}\cr\approx \mathstrut & 3647.45891571999 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^15 - x^14 - 138*x^13 - 80*x^12 + 6278*x^11 + 13450*x^10 - 98056*x^9 - 360148*x^8 + 54921*x^7 + 1300271*x^6 + 477210*x^5 - 1783924*x^4 - 627480*x^3 + 972256*x^2 + 154496*x - 141824)
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^15 - x^14 - 138*x^13 - 80*x^12 + 6278*x^11 + 13450*x^10 - 98056*x^9 - 360148*x^8 + 54921*x^7 + 1300271*x^6 + 477210*x^5 - 1783924*x^4 - 627480*x^3 + 972256*x^2 + 154496*x - 141824, 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^15 - x^14 - 138*x^13 - 80*x^12 + 6278*x^11 + 13450*x^10 - 98056*x^9 - 360148*x^8 + 54921*x^7 + 1300271*x^6 + 477210*x^5 - 1783924*x^4 - 627480*x^3 + 972256*x^2 + 154496*x - 141824);
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^15 - x^14 - 138*x^13 - 80*x^12 + 6278*x^11 + 13450*x^10 - 98056*x^9 - 360148*x^8 + 54921*x^7 + 1300271*x^6 + 477210*x^5 - 1783924*x^4 - 627480*x^3 + 972256*x^2 + 154496*x - 141824);
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 cyclic group of order 15 |
| The 15 conjugacy class representatives for $C_{15}$ |
| Character table for $C_{15}$ |
Intermediate fields
| 3.3.961.1, 5.5.13521270961.1 |
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 | ${\href{/padicField/2.1.0.1}{1} }^{15}$ | $15$ | $15$ | ${\href{/padicField/7.3.0.1}{3} }^{5}$ | R | $15$ | $15$ | $15$ | ${\href{/padicField/23.5.0.1}{5} }^{3}$ | ${\href{/padicField/29.5.0.1}{5} }^{3}$ | R | $15$ | $15$ | $15$ | ${\href{/padicField/47.5.0.1}{5} }^{3}$ | $15$ | $15$ |
In the table, R denotes a ramified prime. 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 |
|---|---|---|---|---|---|---|---|
|
\(11\)
| 11.15.12.4 | $x^{15} - 396 x^{10} + 98978 x^{5} + 74617191$ | $5$ | $3$ | $12$ | $C_{15}$ | $[\ ]_{5}^{3}$ |
|
\(31\)
| 31.15.14.13 | $x^{15} + 465$ | $15$ | $1$ | $14$ | $C_{15}$ | $[\ ]_{15}$ |