Properties

Label 25.1.125...521.1
Degree $25$
Signature $[1, 12]$
Discriminant $1.259\times 10^{49}$
Root discriminant $92.05$
Ramified primes $11, 227$
Class number $5$ (GRH)
Class group $[5]$ (GRH)
Galois group $D_{25}$ (as 25T4)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^25 - 6*x^24 - 11*x^23 + 373*x^22 - 2123*x^21 + 6909*x^20 - 14504*x^19 + 11689*x^18 + 26637*x^17 - 134028*x^16 + 212769*x^15 - 8645*x^14 - 228977*x^13 + 2110377*x^12 + 2314970*x^11 + 6611519*x^10 + 25497478*x^9 + 45079364*x^8 + 81358720*x^7 + 135641936*x^6 + 134356064*x^5 + 105811904*x^4 + 91282176*x^3 + 70338560*x^2 + 75878400*x + 47071232)
 
gp: K = bnfinit(x^25 - 6*x^24 - 11*x^23 + 373*x^22 - 2123*x^21 + 6909*x^20 - 14504*x^19 + 11689*x^18 + 26637*x^17 - 134028*x^16 + 212769*x^15 - 8645*x^14 - 228977*x^13 + 2110377*x^12 + 2314970*x^11 + 6611519*x^10 + 25497478*x^9 + 45079364*x^8 + 81358720*x^7 + 135641936*x^6 + 134356064*x^5 + 105811904*x^4 + 91282176*x^3 + 70338560*x^2 + 75878400*x + 47071232, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![47071232, 75878400, 70338560, 91282176, 105811904, 134356064, 135641936, 81358720, 45079364, 25497478, 6611519, 2314970, 2110377, -228977, -8645, 212769, -134028, 26637, 11689, -14504, 6909, -2123, 373, -11, -6, 1]);
 

\( x^{25} - 6 x^{24} - 11 x^{23} + 373 x^{22} - 2123 x^{21} + 6909 x^{20} - 14504 x^{19} + 11689 x^{18} + 26637 x^{17} - 134028 x^{16} + 212769 x^{15} - 8645 x^{14} - 228977 x^{13} + 2110377 x^{12} + 2314970 x^{11} + 6611519 x^{10} + 25497478 x^{9} + 45079364 x^{8} + 81358720 x^{7} + 135641936 x^{6} + 134356064 x^{5} + 105811904 x^{4} + 91282176 x^{3} + 70338560 x^{2} + 75878400 x + 47071232 \)

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 

Invariants

Degree:  $25$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[1, 12]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(12594008714591324159904490763426952893783035367521\)\(\medspace = 11^{20}\cdot 227^{12}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $92.05$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $11, 227$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $1$
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{5} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{3}$, $\frac{1}{8} a^{10} - \frac{1}{8} a^{9} - \frac{1}{8} a^{4} + \frac{1}{8} a^{3}$, $\frac{1}{16} a^{11} - \frac{1}{16} a^{10} - \frac{1}{8} a^{8} - \frac{1}{8} a^{7} - \frac{3}{16} a^{5} - \frac{1}{16} a^{4}$, $\frac{1}{16} a^{12} - \frac{1}{16} a^{10} - \frac{1}{8} a^{9} - \frac{1}{8} a^{7} + \frac{1}{16} a^{6} - \frac{1}{16} a^{4} + \frac{1}{4} a^{3}$, $\frac{1}{16} a^{13} - \frac{1}{16} a^{10} - \frac{1}{8} a^{9} - \frac{1}{16} a^{7} - \frac{1}{4} a^{5} + \frac{1}{16} a^{4} + \frac{1}{8} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{160} a^{14} + \frac{1}{160} a^{13} + \frac{1}{40} a^{12} - \frac{3}{160} a^{11} + \frac{1}{160} a^{10} + \frac{3}{40} a^{9} + \frac{7}{160} a^{8} - \frac{17}{160} a^{7} - \frac{1}{8} a^{6} + \frac{19}{160} a^{5} + \frac{7}{160} a^{4} + \frac{1}{40} a^{3} + \frac{1}{4} a^{2} - \frac{1}{20} a + \frac{2}{5}$, $\frac{1}{160} a^{15} + \frac{3}{160} a^{13} + \frac{3}{160} a^{12} + \frac{1}{40} a^{11} + \frac{1}{160} a^{10} + \frac{3}{32} a^{9} + \frac{1}{10} a^{8} + \frac{17}{160} a^{7} + \frac{9}{160} a^{6} - \frac{3}{40} a^{5} + \frac{27}{160} a^{4} - \frac{1}{40} a^{3} + \frac{9}{20} a^{2} + \frac{9}{20} a - \frac{2}{5}$, $\frac{1}{320} a^{16} - \frac{1}{320} a^{14} - \frac{1}{320} a^{13} + \frac{1}{40} a^{12} - \frac{7}{320} a^{11} + \frac{11}{320} a^{10} - \frac{1}{10} a^{9} + \frac{29}{320} a^{8} - \frac{3}{320} a^{7} + \frac{1}{40} a^{6} - \frac{69}{320} a^{5} + \frac{3}{20} a^{4} - \frac{9}{20} a^{3} + \frac{19}{40} a^{2} + \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{320} a^{17} - \frac{1}{320} a^{15} - \frac{1}{320} a^{14} + \frac{1}{40} a^{13} - \frac{7}{320} a^{12} - \frac{9}{320} a^{11} - \frac{3}{80} a^{10} + \frac{29}{320} a^{9} + \frac{37}{320} a^{8} - \frac{1}{10} a^{7} + \frac{11}{320} a^{6} - \frac{13}{80} a^{5} - \frac{11}{80} a^{4} - \frac{11}{40} a^{3} - \frac{1}{10} a^{2} - \frac{3}{10} a$, $\frac{1}{1280} a^{18} + \frac{1}{1280} a^{17} + \frac{1}{1280} a^{16} + \frac{1}{640} a^{15} + \frac{1}{1280} a^{14} + \frac{27}{1280} a^{13} - \frac{3}{160} a^{12} + \frac{13}{1280} a^{11} - \frac{21}{1280} a^{10} + \frac{67}{640} a^{9} + \frac{139}{1280} a^{8} + \frac{9}{1280} a^{7} - \frac{9}{1280} a^{6} - \frac{89}{640} a^{5} + \frac{23}{160} a^{4} - \frac{1}{32} a^{3} - \frac{31}{80} a^{2} + \frac{1}{4} a + \frac{1}{5}$, $\frac{1}{1280} a^{19} + \frac{1}{1280} a^{16} - \frac{1}{1280} a^{15} + \frac{1}{640} a^{14} + \frac{1}{256} a^{13} + \frac{21}{1280} a^{12} + \frac{19}{640} a^{11} - \frac{29}{1280} a^{10} + \frac{37}{1280} a^{9} + \frac{11}{640} a^{8} + \frac{15}{128} a^{7} + \frac{71}{1280} a^{6} - \frac{47}{640} a^{5} + \frac{31}{160} a^{4} + \frac{31}{160} a^{3} - \frac{29}{80} a^{2} - \frac{2}{5} a - \frac{2}{5}$, $\frac{1}{33280} a^{20} + \frac{1}{3328} a^{19} + \frac{1}{3328} a^{18} + \frac{3}{33280} a^{17} + \frac{27}{33280} a^{16} + \frac{3}{8320} a^{15} - \frac{1}{6656} a^{14} + \frac{73}{6656} a^{13} - \frac{77}{4160} a^{12} + \frac{165}{6656} a^{11} + \frac{2073}{33280} a^{10} + \frac{681}{8320} a^{9} + \frac{181}{4160} a^{8} - \frac{1243}{33280} a^{7} - \frac{125}{3328} a^{6} - \frac{379}{4160} a^{5} - \frac{539}{4160} a^{4} - \frac{179}{416} a^{3} + \frac{29}{130} a^{2} + \frac{17}{65} a - \frac{1}{5}$, $\frac{1}{1131520} a^{21} - \frac{3}{1131520} a^{20} - \frac{69}{282880} a^{19} - \frac{231}{1131520} a^{18} + \frac{15}{56576} a^{17} - \frac{599}{1131520} a^{16} + \frac{623}{226304} a^{15} - \frac{165}{113152} a^{14} - \frac{16437}{1131520} a^{13} - \frac{19819}{1131520} a^{12} + \frac{801}{282880} a^{11} - \frac{67853}{1131520} a^{10} - \frac{9413}{141440} a^{9} + \frac{729}{226304} a^{8} - \frac{9207}{226304} a^{7} + \frac{3511}{113152} a^{6} + \frac{12331}{141440} a^{5} + \frac{3581}{28288} a^{4} - \frac{3971}{14144} a^{3} + \frac{3529}{17680} a^{2} + \frac{15}{34} a + \frac{2}{5}$, $\frac{1}{2263040} a^{22} - \frac{1}{2263040} a^{21} - \frac{1}{226304} a^{20} - \frac{11}{34816} a^{19} - \frac{47}{1131520} a^{18} + \frac{1701}{2263040} a^{17} - \frac{623}{452608} a^{16} - \frac{677}{282880} a^{15} + \frac{163}{133120} a^{14} + \frac{831}{452608} a^{13} + \frac{13369}{1131520} a^{12} + \frac{227}{174080} a^{11} - \frac{1457}{87040} a^{10} + \frac{136937}{2263040} a^{9} + \frac{80187}{2263040} a^{8} - \frac{4769}{113152} a^{7} - \frac{3937}{43520} a^{6} + \frac{12851}{282880} a^{5} + \frac{2071}{35360} a^{4} + \frac{5893}{14144} a^{3} - \frac{5}{16} a^{2} - \frac{409}{884} a + \frac{1}{5}$, $\frac{1}{5888905318400} a^{23} - \frac{3361}{79579801600} a^{22} + \frac{175391}{452992716800} a^{21} - \frac{83656509}{5888905318400} a^{20} - \frac{28253707}{235556212736} a^{19} + \frac{1663711991}{5888905318400} a^{18} + \frac{441579711}{736113164800} a^{17} + \frac{3364323687}{5888905318400} a^{16} + \frac{15881553247}{5888905318400} a^{15} + \frac{5965315}{29444526592} a^{14} - \frac{104090684557}{5888905318400} a^{13} - \frac{60131789599}{5888905318400} a^{12} + \frac{100165439431}{5888905318400} a^{11} - \frac{17016933613}{452992716800} a^{10} - \frac{202683692863}{2944452659200} a^{9} + \frac{699687738749}{5888905318400} a^{8} - \frac{48250215379}{1472226329600} a^{7} + \frac{1043706611}{17320309760} a^{6} - \frac{159530451189}{736113164800} a^{5} + \frac{2209749501}{11501768200} a^{4} - \frac{4438848533}{184028291200} a^{3} - \frac{5859437349}{23003536400} a^{2} - \frac{67571327}{442375700} a + \frac{294901}{1000850}$, $\frac{1}{352455363221100085074638839911961385718517713295761817600} a^{24} - \frac{13373934906629598928991686853714994179338253}{176227681610550042537319419955980692859258856647880908800} a^{23} + \frac{58738350182265315318454092990042132103159431766261}{352455363221100085074638839911961385718517713295761817600} a^{22} + \frac{15859406045756503774360437239338421085442960065013}{70491072644220017014927767982392277143703542659152363520} a^{21} - \frac{412544828916601303625344640264706943194348801626847}{352455363221100085074638839911961385718517713295761817600} a^{20} - \frac{121383436949170206700131954596484682279756167976520639}{352455363221100085074638839911961385718517713295761817600} a^{19} - \frac{14111035880437268559111406694786699015131614398701751}{88113840805275021268659709977990346429629428323940454400} a^{18} - \frac{17814740491789201881153335235569085334863371874129643}{27111951017007698851895295377843183516809054868904755200} a^{17} - \frac{216173588425436127009684322844386465985083760900549087}{352455363221100085074638839911961385718517713295761817600} a^{16} + \frac{46838012093131429540132691919904460068353116514816271}{22028460201318755317164927494497586607407357080985113600} a^{15} + \frac{146352821325732189438973931062147404335339365164137993}{352455363221100085074638839911961385718517713295761817600} a^{14} + \frac{1268555122076849164646288813153137821541783914576011283}{70491072644220017014927767982392277143703542659152363520} a^{13} + \frac{10409217555196610735366874252376706928565392551348778179}{352455363221100085074638839911961385718517713295761817600} a^{12} - \frac{9615362481485718923736549999916653710564567602956002091}{352455363221100085074638839911961385718517713295761817600} a^{11} - \frac{76694500680016180512688836797339223987137707990660737}{4762910313798649798305930269080559266466455585077862400} a^{10} - \frac{196846866240691621966191702012989510311919925657300063}{1580517323861435359079097936824938949410393333164851200} a^{9} - \frac{2201682425836955832734840097700310816125463062201875367}{176227681610550042537319419955980692859258856647880908800} a^{8} + \frac{10061008632037122503042419507656296298553217720477016963}{88113840805275021268659709977990346429629428323940454400} a^{7} - \frac{159355251545272496984580431971413302750839998546920147}{22028460201318755317164927494497586607407357080985113600} a^{6} + \frac{5378118884662697887577272828609097618560652367374807461}{22028460201318755317164927494497586607407357080985113600} a^{5} - \frac{9249212198722519052683320553313179744430017574164847}{129579177654816207748028985261750509455337394594030080} a^{4} - \frac{1827327200048218699720733313856323413280645562587329163}{5507115050329688829291231873624396651851839270246278400} a^{3} + \frac{229033112183865814445041783880299448843702132165397567}{688389381291211103661403984203049581481479908780784800} a^{2} - \frac{822283980695115954298499172904314639114284234122984}{1654782166565411306878374962026561493945865165338425} a - \frac{16649086205137713394940185664684580501549901211}{55361937975273251541167268993285820424582101700}$

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 

Class group and class number

$C_{5}$, which has order $5$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $12$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 3887937729197100.0 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{1}\cdot(2\pi)^{12}\cdot 3887937729197100.0 \cdot 5}{2\sqrt{12594008714591324159904490763426952893783035367521}}\approx 20.7379455745518$ (assuming GRH)

Galois group

$D_{25}$ (as 25T4):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 50
The 14 conjugacy class representatives for $D_{25}$
Character table for $D_{25}$

Intermediate fields

5.1.51529.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/LocalNumberField/2.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }$ $25$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ $25$ R ${\href{/LocalNumberField/13.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ $25$ $25$ $25$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ $25$ $25$ $25$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{5}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 
magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
11Data not computed
227Data not computed

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.227.2t1.a.a$1$ $ 227 $ \(\Q(\sqrt{-227}) \) $C_2$ (as 2T1) $1$ $-1$
* 2.227.5t2.a.b$2$ $ 227 $ 5.1.51529.1 $D_{5}$ (as 5T2) $1$ $0$
* 2.227.5t2.a.a$2$ $ 227 $ 5.1.51529.1 $D_{5}$ (as 5T2) $1$ $0$
* 2.27467.25t4.a.f$2$ $ 11^{2} \cdot 227 $ 25.1.12594008714591324159904490763426952893783035367521.1 $D_{25}$ (as 25T4) $1$ $0$
* 2.27467.25t4.a.d$2$ $ 11^{2} \cdot 227 $ 25.1.12594008714591324159904490763426952893783035367521.1 $D_{25}$ (as 25T4) $1$ $0$
* 2.27467.25t4.a.h$2$ $ 11^{2} \cdot 227 $ 25.1.12594008714591324159904490763426952893783035367521.1 $D_{25}$ (as 25T4) $1$ $0$
* 2.27467.25t4.a.c$2$ $ 11^{2} \cdot 227 $ 25.1.12594008714591324159904490763426952893783035367521.1 $D_{25}$ (as 25T4) $1$ $0$
* 2.27467.25t4.a.i$2$ $ 11^{2} \cdot 227 $ 25.1.12594008714591324159904490763426952893783035367521.1 $D_{25}$ (as 25T4) $1$ $0$
* 2.27467.25t4.a.g$2$ $ 11^{2} \cdot 227 $ 25.1.12594008714591324159904490763426952893783035367521.1 $D_{25}$ (as 25T4) $1$ $0$
* 2.27467.25t4.a.a$2$ $ 11^{2} \cdot 227 $ 25.1.12594008714591324159904490763426952893783035367521.1 $D_{25}$ (as 25T4) $1$ $0$
* 2.27467.25t4.a.j$2$ $ 11^{2} \cdot 227 $ 25.1.12594008714591324159904490763426952893783035367521.1 $D_{25}$ (as 25T4) $1$ $0$
* 2.27467.25t4.a.e$2$ $ 11^{2} \cdot 227 $ 25.1.12594008714591324159904490763426952893783035367521.1 $D_{25}$ (as 25T4) $1$ $0$
* 2.27467.25t4.a.b$2$ $ 11^{2} \cdot 227 $ 25.1.12594008714591324159904490763426952893783035367521.1 $D_{25}$ (as 25T4) $1$ $0$

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 *.