Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^25 - 150*x^20 - 35*x^15 + 795*x^10 - 365*x^5 - 1)
gp: K = bnfinit(y^25 - 150*y^20 - 35*y^15 + 795*y^10 - 365*y^5 - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^25 - 150*x^20 - 35*x^15 + 795*x^10 - 365*x^5 - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^25 - 150*x^20 - 35*x^15 + 795*x^10 - 365*x^5 - 1)
\( x^{25} - 150x^{20} - 35x^{15} + 795x^{10} - 365x^{5} - 1 \)
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: | $[5, 10]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(2710505431213761085018632002174854278564453125\)
\(\medspace = 5^{65}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(65.66\) | 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: | not computed | ||
| Ramified primes: |
\(5\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{5}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{39188093}a^{20}-\frac{5008502}{39188093}a^{15}+\frac{3491276}{39188093}a^{10}-\frac{7980222}{39188093}a^{5}-\frac{18484549}{39188093}$, $\frac{1}{39188093}a^{21}-\frac{5008502}{39188093}a^{16}+\frac{3491276}{39188093}a^{11}-\frac{7980222}{39188093}a^{6}-\frac{18484549}{39188093}a$, $\frac{1}{39188093}a^{22}-\frac{5008502}{39188093}a^{17}+\frac{3491276}{39188093}a^{12}-\frac{7980222}{39188093}a^{7}-\frac{18484549}{39188093}a^{2}$, $\frac{1}{39188093}a^{23}-\frac{5008502}{39188093}a^{18}+\frac{3491276}{39188093}a^{13}-\frac{7980222}{39188093}a^{8}-\frac{18484549}{39188093}a^{3}$, $\frac{1}{39188093}a^{24}-\frac{5008502}{39188093}a^{19}+\frac{3491276}{39188093}a^{14}-\frac{7980222}{39188093}a^{9}-\frac{18484549}{39188093}a^{4}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (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: |
$\frac{3308892}{39188093}a^{22}+\frac{496303386}{39188093}a^{17}+\frac{120273557}{39188093}a^{12}-\frac{2614505374}{39188093}a^{7}+\frac{1187169446}{39188093}a^{2}$, $a$, $\frac{12298414}{39188093}a^{24}+\frac{1844616939}{39188093}a^{19}+\frac{452101976}{39188093}a^{14}-\frac{9754747797}{39188093}a^{9}+\frac{4388333493}{39188093}a^{4}$, $\frac{816875}{39188093}a^{21}-\frac{122350143}{39188093}a^{16}-\frac{55761761}{39188093}a^{11}+\frac{644057602}{39188093}a^{6}-\frac{197791010}{39188093}a$, $\frac{30198571}{39188093}a^{24}+\frac{35062326}{39188093}a^{23}+\frac{4367616}{39188093}a^{22}-\frac{72718}{799757}a^{21}-\frac{641300}{39188093}a^{20}-\frac{4529900397}{39188093}a^{19}-\frac{5259560421}{39188093}a^{18}-\frac{655087190}{39188093}a^{17}+\frac{10906991}{799757}a^{16}+\frac{96230320}{39188093}a^{15}-\frac{1039754836}{39188093}a^{14}-\frac{1195547937}{39188093}a^{13}-\frac{161077772}{39188093}a^{12}+\frac{2651968}{799757}a^{11}+\frac{17206662}{39188093}a^{10}+\frac{24014291512}{39188093}a^{9}+\frac{27895970959}{39188093}a^{8}+\frac{3459373213}{39188093}a^{7}-\frac{57877579}{799757}a^{6}-\frac{522893851}{39188093}a^{5}-\frac{11102489898}{39188093}a^{4}-\frac{12957881582}{39188093}a^{3}-\frac{1551318768}{39188093}a^{2}+\frac{25438936}{799757}a+\frac{252586409}{39188093}$, $\frac{182703606}{39188093}a^{24}+\frac{43234973}{39188093}a^{23}+\frac{3805031}{39188093}a^{22}-\frac{1995878}{39188093}a^{21}-\frac{168429}{5598299}a^{20}-\frac{27405915469}{39188093}a^{19}-\frac{6485523831}{39188093}a^{18}-\frac{570876220}{39188093}a^{17}+\frac{299380409}{39188093}a^{16}+\frac{25290038}{5598299}a^{15}-\frac{6338240575}{39188093}a^{14}-\frac{1471614595}{39188093}a^{13}-\frac{114988886}{39188093}a^{12}+\frac{69796467}{39188093}a^{11}+\frac{2004958}{5598299}a^{10}+\frac{145232270136}{39188093}a^{9}+\frac{34392155780}{39188093}a^{8}+\frac{3036811608}{39188093}a^{7}-\frac{1548141538}{39188093}a^{6}-\frac{129154848}{5598299}a^{5}-\frac{66967528284}{39188093}a^{4}-\frac{15961254619}{39188093}a^{3}-\frac{1530430455}{39188093}a^{2}+\frac{685305520}{39188093}a+\frac{69644930}{5598299}$, $\frac{1148761}{39188093}a^{24}-\frac{3051222}{39188093}a^{23}-\frac{3656599}{39188093}a^{22}-\frac{347598}{39188093}a^{21}-\frac{543493}{39188093}a^{20}+\frac{171892227}{39188093}a^{19}+\frac{458683629}{39188093}a^{18}+\frac{547031566}{39188093}a^{17}+\frac{53434764}{39188093}a^{16}+\frac{80837706}{39188093}a^{15}+\frac{102857142}{39188093}a^{14}-\frac{41254803}{39188093}a^{13}+\frac{343854844}{39188093}a^{12}-\frac{179631489}{39188093}a^{11}+\frac{120960271}{39188093}a^{10}-\frac{804116687}{39188093}a^{9}-\frac{2757444590}{39188093}a^{8}-\frac{2430815006}{39188093}a^{7}-\frac{686153830}{39188093}a^{6}-\frac{260902073}{39188093}a^{5}+\frac{299989832}{39188093}a^{4}+\frac{1311816115}{39188093}a^{3}+\frac{1037551380}{39188093}a^{2}+\frac{343604045}{39188093}a+\frac{41844363}{39188093}$, $\frac{45927447}{39188093}a^{24}-\frac{419189}{5598299}a^{23}-\frac{1379460}{39188093}a^{22}+\frac{949595}{39188093}a^{21}-\frac{562585}{39188093}a^{20}+\frac{6889007921}{39188093}a^{19}+\frac{62845393}{5598299}a^{18}+\frac{206561113}{39188093}a^{17}-\frac{142302117}{39188093}a^{16}+\frac{84210970}{39188093}a^{15}+\frac{1623942332}{39188093}a^{14}+\frac{19624313}{5598299}a^{13}+\frac{101850647}{39188093}a^{12}-\frac{53622673}{39188093}a^{11}+\frac{46088886}{39188093}a^{10}-\frac{36525138823}{39188093}a^{9}-\frac{333399040}{5598299}a^{8}-\frac{1066619207}{39188093}a^{7}+\frac{723959459}{39188093}a^{6}-\frac{422561605}{39188093}a^{5}+\frac{16642609775}{39188093}a^{4}+\frac{115503625}{5598299}a^{3}+\frac{140303137}{39188093}a^{2}-\frac{410076769}{39188093}a-\frac{96675966}{39188093}$, $\frac{18995085}{39188093}a^{24}-\frac{3509349}{39188093}a^{23}-\frac{2329055}{39188093}a^{22}-\frac{198843}{5598299}a^{21}-\frac{725719}{39188093}a^{20}+\frac{2848567359}{39188093}a^{19}+\frac{525834233}{39188093}a^{18}+\frac{348863230}{39188093}a^{17}+\frac{29752375}{5598299}a^{16}+\frac{108625281}{39188093}a^{15}+\frac{768653068}{39188093}a^{14}+\frac{208089498}{39188093}a^{13}+\frac{155473320}{39188093}a^{12}+\frac{18068724}{5598299}a^{11}+\frac{60013564}{39188093}a^{10}-\frac{15004699676}{39188093}a^{9}-\frac{2777040645}{39188093}a^{8}-\frac{1790578666}{39188093}a^{7}-\frac{155329281}{5598299}a^{6}-\frac{523039596}{39188093}a^{5}+\frac{6450844074}{39188093}a^{4}+\frac{918334259}{39188093}a^{3}+\frac{511191813}{39188093}a^{2}+\frac{32746244}{5598299}a-\frac{5263378}{39188093}$, $\frac{45267055}{39188093}a^{24}-\frac{17953990}{39188093}a^{23}-\frac{1004264}{5598299}a^{22}-\frac{187389}{5598299}a^{21}-\frac{45635}{5598299}a^{20}+\frac{6789404337}{39188093}a^{19}+\frac{2693891784}{39188093}a^{18}+\frac{150490463}{5598299}a^{17}+\frac{28140320}{5598299}a^{16}+\frac{6833796}{5598299}a^{15}+\frac{1681607055}{39188093}a^{14}+\frac{509509794}{39188093}a^{13}+\frac{57461135}{5598299}a^{12}+\frac{1699374}{5598299}a^{11}+\frac{3209280}{5598299}a^{10}-\frac{35840585440}{39188093}a^{9}-\frac{14318504441}{39188093}a^{8}-\frac{784627001}{5598299}a^{7}-\frac{140569399}{5598299}a^{6}-\frac{19910475}{5598299}a^{5}+\frac{16058286579}{39188093}a^{4}+\frac{7238270475}{39188093}a^{3}+\frac{286565879}{5598299}a^{2}+\frac{99971467}{5598299}a+\frac{13093491}{5598299}$, $\frac{22921686}{39188093}a^{24}-\frac{111276}{911351}a^{23}+\frac{2329055}{39188093}a^{22}+\frac{199166}{39188093}a^{21}+\frac{858439}{39188093}a^{20}-\frac{3437630778}{39188093}a^{19}+\frac{16705032}{911351}a^{18}-\frac{348863230}{39188093}a^{17}-\frac{29590110}{39188093}a^{16}-\frac{128577255}{39188093}a^{15}-\frac{895453917}{39188093}a^{14}+\frac{1855561}{911351}a^{13}-\frac{155473320}{39188093}a^{12}-\frac{49234469}{39188093}a^{11}-\frac{57874476}{39188093}a^{10}+\frac{18183105424}{39188093}a^{9}-\frac{89784612}{911351}a^{8}+\frac{1790578666}{39188093}a^{7}+\frac{80157228}{39188093}a^{6}+\frac{602941453}{39188093}a^{5}-\frac{7985079560}{39188093}a^{4}+\frac{49489114}{911351}a^{3}-\frac{511191813}{39188093}a^{2}+\frac{227651216}{39188093}a-\frac{50270009}{39188093}$, $\frac{28142381}{39188093}a^{24}-\frac{6406177}{39188093}a^{23}-\frac{2935614}{39188093}a^{22}-\frac{1457056}{39188093}a^{21}+\frac{30242}{39188093}a^{20}+\frac{4219568208}{39188093}a^{19}+\frac{961311150}{39188093}a^{18}+\frac{439858488}{39188093}a^{17}+\frac{217964024}{39188093}a^{16}-\frac{5138039}{39188093}a^{15}+\frac{1252489443}{39188093}a^{14}+\frac{166682224}{39188093}a^{13}+\frac{175951663}{39188093}a^{12}+\frac{139273153}{39188093}a^{11}+\frac{88822436}{39188093}a^{10}-\frac{22186824361}{39188093}a^{9}-\frac{5128167760}{39188093}a^{8}-\frac{2416171323}{39188093}a^{7}-\frac{1005182295}{39188093}a^{6}+\frac{99967249}{39188093}a^{5}+\frac{9106727685}{39188093}a^{4}+\frac{2395505258}{39188093}a^{3}+\frac{649777125}{39188093}a^{2}-\frac{12776924}{39188093}a-\frac{30772306}{39188093}$, $\frac{4913518}{39188093}a^{24}+\frac{95570}{39188093}a^{23}-\frac{11416211}{39188093}a^{22}-\frac{7611008}{39188093}a^{21}-\frac{6086387}{39188093}a^{20}+\frac{731473570}{39188093}a^{19}-\frac{19168238}{39188093}a^{18}+\frac{1708236597}{39188093}a^{17}+\frac{1138036079}{39188093}a^{16}+\frac{909817480}{39188093}a^{15}+\frac{1002379828}{39188093}a^{14}+\frac{719209192}{39188093}a^{13}+\frac{1026726971}{39188093}a^{12}+\frac{806851883}{39188093}a^{11}+\frac{682509903}{39188093}a^{10}-\frac{3306205128}{39188093}a^{9}+\frac{596670821}{39188093}a^{8}-\frac{8613986623}{39188093}a^{7}-\frac{5656566130}{39188093}a^{6}-\frac{4489359306}{39188093}a^{5}-\frac{2314041276}{39188093}a^{4}-\frac{3613608139}{39188093}a^{3}+\frac{1077963766}{39188093}a^{2}+\frac{91059718}{39188093}a-\frac{115321191}{39188093}$, $\frac{50511858}{39188093}a^{24}+\frac{3828794}{39188093}a^{23}+\frac{4204654}{39188093}a^{22}-\frac{296973}{39188093}a^{21}+\frac{599908}{39188093}a^{20}+\frac{7577793450}{39188093}a^{19}-\frac{573670805}{39188093}a^{18}-\frac{629997177}{39188093}a^{17}+\frac{44982724}{39188093}a^{16}-\frac{89327506}{39188093}a^{15}+\frac{1615915514}{39188093}a^{14}-\frac{230554458}{39188093}a^{13}-\frac{252039296}{39188093}a^{12}-\frac{54519140}{39188093}a^{11}-\frac{119980149}{39188093}a^{10}-\frac{40223341868}{39188093}a^{9}+\frac{2916413970}{39188093}a^{8}+\frac{3279504107}{39188093}a^{7}-\frac{342149006}{39188093}a^{6}+\frac{484618885}{39188093}a^{5}+\frac{19078664957}{39188093}a^{4}-\frac{892424417}{39188093}a^{3}-\frac{1112320238}{39188093}a^{2}+\frac{492536039}{39188093}a+\frac{65042811}{39188093}$
| 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: | \( 13680882102054.447 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{5}\cdot(2\pi)^{10}\cdot 13680882102054.447 \cdot 1}{2\cdot\sqrt{2710505431213761085018632002174854278564453125}}\cr\approx \mathstrut & 0.403188025798417 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^25 - 150*x^20 - 35*x^15 + 795*x^10 - 365*x^5 - 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 - 150*x^20 - 35*x^15 + 795*x^10 - 365*x^5 - 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 - 150*x^20 - 35*x^15 + 795*x^10 - 365*x^5 - 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 - 150*x^20 - 35*x^15 + 795*x^10 - 365*x^5 - 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
$C_5^4:C_{20}$ (as 25T94):
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 12500 |
| The 56 conjugacy class representatives for $C_5^4:C_{20}$ |
| Character table for $C_5^4:C_{20}$ |
Intermediate fields
| 5.5.390625.1 |
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]
Sibling fields
| Degree 25 siblings: | data not computed |
| Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | $20{,}\,{\href{/padicField/2.5.0.1}{5} }$ | $20{,}\,{\href{/padicField/3.5.0.1}{5} }$ | R | ${\href{/padicField/7.4.0.1}{4} }^{5}{,}\,{\href{/padicField/7.1.0.1}{1} }^{5}$ | ${\href{/padicField/11.5.0.1}{5} }^{5}$ | $20{,}\,{\href{/padicField/13.5.0.1}{5} }$ | $20{,}\,{\href{/padicField/17.5.0.1}{5} }$ | ${\href{/padicField/19.10.0.1}{10} }^{2}{,}\,{\href{/padicField/19.5.0.1}{5} }$ | $20{,}\,{\href{/padicField/23.5.0.1}{5} }$ | ${\href{/padicField/29.10.0.1}{10} }^{2}{,}\,{\href{/padicField/29.5.0.1}{5} }$ | ${\href{/padicField/31.5.0.1}{5} }^{5}$ | $20{,}\,{\href{/padicField/37.5.0.1}{5} }$ | ${\href{/padicField/41.5.0.1}{5} }^{5}$ | ${\href{/padicField/43.4.0.1}{4} }^{5}{,}\,{\href{/padicField/43.1.0.1}{1} }^{5}$ | $20{,}\,{\href{/padicField/47.5.0.1}{5} }$ | $20{,}\,{\href{/padicField/53.5.0.1}{5} }$ | ${\href{/padicField/59.10.0.1}{10} }^{2}{,}\,{\href{/padicField/59.5.0.1}{5} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| Deg $25$ | $25$ | $1$ | $65$ |