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Properties

Label 25.25.2014772834...0625.1
Degree $25$
Signature $[25, 0]$
Discriminant $5^{40}\cdot 131^{20}$
Root discriminant $648.89$
Ramified primes $5, 131$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_5^2$ (as 25T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3793733257, -184088651755, 889938263265, 2568415257135, -1431049624960, -4591675173046, 1636194190540, 3432829509630, -1392866509160, -1156119349540, 602962608969, 148895760435, -113804899720, -4512083030, 10672153285, -560170703, -536477370, 54719040, 14802975, -1972695, -221956, 34860, 1680, -300, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^25 - 5*x^24 - 300*x^23 + 1680*x^22 + 34860*x^21 - 221956*x^20 - 1972695*x^19 + 14802975*x^18 + 54719040*x^17 - 536477370*x^16 - 560170703*x^15 + 10672153285*x^14 - 4512083030*x^13 - 113804899720*x^12 + 148895760435*x^11 + 602962608969*x^10 - 1156119349540*x^9 - 1392866509160*x^8 + 3432829509630*x^7 + 1636194190540*x^6 - 4591675173046*x^5 - 1431049624960*x^4 + 2568415257135*x^3 + 889938263265*x^2 - 184088651755*x + 3793733257)
 
gp: K = bnfinit(x^25 - 5*x^24 - 300*x^23 + 1680*x^22 + 34860*x^21 - 221956*x^20 - 1972695*x^19 + 14802975*x^18 + 54719040*x^17 - 536477370*x^16 - 560170703*x^15 + 10672153285*x^14 - 4512083030*x^13 - 113804899720*x^12 + 148895760435*x^11 + 602962608969*x^10 - 1156119349540*x^9 - 1392866509160*x^8 + 3432829509630*x^7 + 1636194190540*x^6 - 4591675173046*x^5 - 1431049624960*x^4 + 2568415257135*x^3 + 889938263265*x^2 - 184088651755*x + 3793733257, 1)
 

Normalized defining polynomial

\( x^{25} - 5 x^{24} - 300 x^{23} + 1680 x^{22} + 34860 x^{21} - 221956 x^{20} - 1972695 x^{19} + 14802975 x^{18} + 54719040 x^{17} - 536477370 x^{16} - 560170703 x^{15} + 10672153285 x^{14} - 4512083030 x^{13} - 113804899720 x^{12} + 148895760435 x^{11} + 602962608969 x^{10} - 1156119349540 x^{9} - 1392866509160 x^{8} + 3432829509630 x^{7} + 1636194190540 x^{6} - 4591675173046 x^{5} - 1431049624960 x^{4} + 2568415257135 x^{3} + 889938263265 x^{2} - 184088651755 x + 3793733257 \)

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

Invariants

Degree:  $25$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[25, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(20147728344624516737564339110181128448344688877114094793796539306640625=5^{40}\cdot 131^{20}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $648.89$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $5, 131$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3275=5^{2}\cdot 131\)
Dirichlet character group:    $\lbrace$$\chi_{3275}(1,·)$, $\chi_{3275}(451,·)$, $\chi_{3275}(716,·)$, $\chi_{3275}(2316,·)$, $\chi_{3275}(1101,·)$, $\chi_{3275}(656,·)$, $\chi_{3275}(1106,·)$, $\chi_{3275}(1006,·)$, $\chi_{3275}(2971,·)$, $\chi_{3275}(1756,·)$, $\chi_{3275}(1661,·)$, $\chi_{3275}(1311,·)$, $\chi_{3275}(1761,·)$, $\chi_{3275}(1371,·)$, $\chi_{3275}(2026,·)$, $\chi_{3275}(2411,·)$, $\chi_{3275}(1966,·)$, $\chi_{3275}(61,·)$, $\chi_{3275}(2416,·)$, $\chi_{3275}(2681,·)$, $\chi_{3275}(3066,·)$, $\chi_{3275}(351,·)$, $\chi_{3275}(2621,·)$, $\chi_{3275}(446,·)$, $\chi_{3275}(3071,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{19} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{1106} a^{20} + \frac{134}{553} a^{19} + \frac{138}{553} a^{18} - \frac{95}{553} a^{17} - \frac{15}{553} a^{16} - \frac{9}{79} a^{15} + \frac{109}{553} a^{14} - \frac{86}{553} a^{13} - \frac{143}{1106} a^{12} + \frac{95}{553} a^{11} - \frac{276}{553} a^{10} - \frac{61}{553} a^{9} - \frac{369}{1106} a^{8} - \frac{14}{79} a^{7} + \frac{407}{1106} a^{6} + \frac{109}{1106} a^{5} + \frac{43}{1106} a^{4} - \frac{178}{553} a^{3} + \frac{228}{553} a^{2} - \frac{5}{553} a - \frac{60}{553}$, $\frac{1}{1106} a^{21} - \frac{211}{1106} a^{19} - \frac{4}{79} a^{18} + \frac{1}{79} a^{17} + \frac{86}{553} a^{16} + \frac{253}{1106} a^{15} + \frac{11}{553} a^{14} + \frac{27}{553} a^{13} - \frac{14}{79} a^{12} - \frac{43}{1106} a^{11} - \frac{195}{553} a^{10} - \frac{150}{553} a^{9} + \frac{131}{553} a^{8} - \frac{153}{1106} a^{7} - \frac{13}{553} a^{6} - \frac{59}{158} a^{5} + \frac{143}{553} a^{4} - \frac{179}{553} a^{3} - \frac{5}{1106} a^{2} + \frac{174}{553} a - \frac{467}{1106}$, $\frac{1}{2212} a^{22} - \frac{1}{2212} a^{20} - \frac{13}{158} a^{19} - \frac{13}{316} a^{18} + \frac{22}{553} a^{17} + \frac{9}{553} a^{16} + \frac{53}{1106} a^{15} + \frac{122}{553} a^{14} + \frac{13}{158} a^{13} - \frac{211}{2212} a^{12} - \frac{153}{1106} a^{11} - \frac{643}{2212} a^{10} - \frac{513}{1106} a^{9} + \frac{165}{1106} a^{8} + \frac{421}{1106} a^{7} + \frac{16}{79} a^{6} + \frac{264}{553} a^{5} + \frac{465}{1106} a^{4} - \frac{55}{1106} a^{3} + \frac{248}{553} a^{2} - \frac{227}{553} a + \frac{113}{316}$, $\frac{1}{4424} a^{23} - \frac{1}{4424} a^{22} + \frac{1}{4424} a^{21} - \frac{1}{4424} a^{20} - \frac{755}{4424} a^{19} + \frac{1083}{4424} a^{18} + \frac{31}{2212} a^{17} - \frac{281}{2212} a^{16} - \frac{389}{2212} a^{15} + \frac{67}{1106} a^{14} - \frac{277}{4424} a^{13} + \frac{317}{4424} a^{12} + \frac{597}{4424} a^{11} - \frac{983}{4424} a^{10} - \frac{162}{553} a^{9} + \frac{1041}{2212} a^{8} + \frac{29}{79} a^{7} + \frac{963}{2212} a^{6} + \frac{1039}{2212} a^{5} + \frac{871}{2212} a^{4} + \frac{195}{553} a^{3} - \frac{837}{2212} a^{2} - \frac{231}{632} a - \frac{1205}{4424}$, $\frac{1}{2972909729556118073729298586246959405678602820222056110240213455514441472358183865511882878929101428232656534368868943519913217552} a^{24} + \frac{65086327553113788946753760887350893405871374120264098599610972003722278267896673160368867010167773405930210196968958557173431}{743227432389029518432324646561739851419650705055514027560053363878610368089545966377970719732275357058164133592217235879978304388} a^{23} + \frac{6713899415127089922798868801158461202806759074991459106646867711434713799060978979851999185631311753561296212425638876090235}{371613716194514759216162323280869925709825352527757013780026681939305184044772983188985359866137678529082066796108617939989152194} a^{22} - \frac{92920754728566777226412593941766933188422309643194906914230743295705710745725899110347910116754049617667729822512490374318243}{371613716194514759216162323280869925709825352527757013780026681939305184044772983188985359866137678529082066796108617939989152194} a^{21} + \frac{236683685625003964449461213258975886661119535665045610668411953951788585476439890891019647998182343457503165169063757977727}{712586224725819289005105126137813855627661270427146718657769284639127869692757398253087938381855567649246532686689583777543916} a^{20} + \frac{13255673934084208586489954064167099979860632493750630314932061553400989808160951604121376146339851166654443616743879697120914619}{371613716194514759216162323280869925709825352527757013780026681939305184044772983188985359866137678529082066796108617939989152194} a^{19} - \frac{529209956233687267784239664652166251637983302451081546177944842239386770861321770671439722637308803874968000546101451586040040127}{2972909729556118073729298586246959405678602820222056110240213455514441472358183865511882878929101428232656534368868943519913217552} a^{18} - \frac{85663625037735857632068616665201649401037929495159538896392644188320370899019174837332855810092927477871005489569225154180794353}{371613716194514759216162323280869925709825352527757013780026681939305184044772983188985359866137678529082066796108617939989152194} a^{17} - \frac{31713319486160722309443446878035549310111424860524816294761136472090744904878251592584769239749915576032540694111049327449756683}{371613716194514759216162323280869925709825352527757013780026681939305184044772983188985359866137678529082066796108617939989152194} a^{16} + \frac{51482583653849657032115111919666339667749215921779890335787469727292593275605162533910673916136823245052017735373033442388899813}{212350694968294148123521327589068528977043058587289722160015246822460105168441704679420205637792959159475466740633495965708086968} a^{15} + \frac{493973389133861631511252370093569734669164293506052581174306883339589873945546243393392366214542113900598562309649199321615598047}{2972909729556118073729298586246959405678602820222056110240213455514441472358183865511882878929101428232656534368868943519913217552} a^{14} - \frac{11289162235444837123918612610622105889034468088706407253073285198126947870847737381608294151985585262990531750596468767264499449}{106175347484147074061760663794534264488521529293644861080007623411230052584220852339710102818896479579737733370316747982854043484} a^{13} + \frac{3612801334028671611197420105831884090256036674912405202926718830980211625162613224019968065543450997895568788257522432140569403}{1486454864778059036864649293123479702839301410111028055120106727757220736179091932755941439464550714116328267184434471759956608776} a^{12} + \frac{176016604065181708573555182670042796959537436642979955419012958409851470176230150763145514008418158257914421943406259607314513167}{1486454864778059036864649293123479702839301410111028055120106727757220736179091932755941439464550714116328267184434471759956608776} a^{11} - \frac{775786049738149204914246586419115807899848527319402518410254498903589055427382664852477589168606868359102409805563631501765854839}{2972909729556118073729298586246959405678602820222056110240213455514441472358183865511882878929101428232656534368868943519913217552} a^{10} + \frac{20902401292011116101809338178888612667160407271410828069434793412158313122416351625034593780380361927618110686940060385796411543}{212350694968294148123521327589068528977043058587289722160015246822460105168441704679420205637792959159475466740633495965708086968} a^{9} - \frac{706609156341512846571855636499566452897061828727433014337145930043805945461701572535426396949796287251836470178069040795742861001}{1486454864778059036864649293123479702839301410111028055120106727757220736179091932755941439464550714116328267184434471759956608776} a^{8} + \frac{386112652352500933772086741604697271939917962518705316231875719794782293327934947973444713642442987176788547489798000395530932555}{1486454864778059036864649293123479702839301410111028055120106727757220736179091932755941439464550714116328267184434471759956608776} a^{7} - \frac{221566899977698571918525606912155998597363536315247743798354445720852785814705899165944221853744693946967880406496578315792396367}{743227432389029518432324646561739851419650705055514027560053363878610368089545966377970719732275357058164133592217235879978304388} a^{6} - \frac{51788598726329771082407491717925204414258838946940443662842432268384418542760532891195850929980716305976787279387653297419472461}{371613716194514759216162323280869925709825352527757013780026681939305184044772983188985359866137678529082066796108617939989152194} a^{5} - \frac{85385906417346065437944489053286335906364559097947758130688994356684453305097574703257412670640104765642731911292130883060687535}{1486454864778059036864649293123479702839301410111028055120106727757220736179091932755941439464550714116328267184434471759956608776} a^{4} - \frac{126332833344235774737234854805391276611470974332678499999618957190930663745161317706072926132629769269853309125123969748572221667}{1486454864778059036864649293123479702839301410111028055120106727757220736179091932755941439464550714116328267184434471759956608776} a^{3} + \frac{1470364766762477643193313310515185116721351106425268540885262568268566412661943378437111155800713480996039731085474156057442883601}{2972909729556118073729298586246959405678602820222056110240213455514441472358183865511882878929101428232656534368868943519913217552} a^{2} + \frac{652541173263315717062130113413742598653712703195599969927839629580698069784949903735316139418506149929657410036206323043455642589}{1486454864778059036864649293123479702839301410111028055120106727757220736179091932755941439464550714116328267184434471759956608776} a - \frac{325297129500354075477439313086660555117448476980543926207512148634728654516029092365736331400415235254085659878621688551565885}{3969171868566245759318155655870439793963421655837191068411499940606730937727882330456452441827905778681784425058570018050618448}$

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

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $24$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
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)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 1865279722043240000000000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_5^2$ (as 25T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 25
The 25 conjugacy class representatives for $C_5^2$
Character table for $C_5^2$ is not computed

Intermediate fields

5.5.390625.1, 5.5.115039031640625.3, 5.5.115039031640625.4, 5.5.115039031640625.5, 5.5.115039031640625.2, 5.5.294499921.1

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 ${\href{/LocalNumberField/2.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/3.5.0.1}{5} }^{5}$ R ${\href{/LocalNumberField/7.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{5}$

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

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];
 
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]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
5Data not computed
131Data not computed