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Properties

Label 25.25.3909933000...6001.1
Degree $25$
Signature $[25, 0]$
Discriminant $251^{24}$
Root discriminant $201.23$
Ramified prime $251$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_{25}$ (as 25T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4768, 23968, -1066000, -2051852, 40531608, 88908581, -263064528, -408815960, 321873492, 510501238, -167117297, -299088119, 42784540, 96923558, -5136393, -18517488, 143339, 2128729, 26182, -146272, -2591, 5789, 87, -120, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^25 - x^24 - 120*x^23 + 87*x^22 + 5789*x^21 - 2591*x^20 - 146272*x^19 + 26182*x^18 + 2128729*x^17 + 143339*x^16 - 18517488*x^15 - 5136393*x^14 + 96923558*x^13 + 42784540*x^12 - 299088119*x^11 - 167117297*x^10 + 510501238*x^9 + 321873492*x^8 - 408815960*x^7 - 263064528*x^6 + 88908581*x^5 + 40531608*x^4 - 2051852*x^3 - 1066000*x^2 + 23968*x + 4768)
 
gp: K = bnfinit(x^25 - x^24 - 120*x^23 + 87*x^22 + 5789*x^21 - 2591*x^20 - 146272*x^19 + 26182*x^18 + 2128729*x^17 + 143339*x^16 - 18517488*x^15 - 5136393*x^14 + 96923558*x^13 + 42784540*x^12 - 299088119*x^11 - 167117297*x^10 + 510501238*x^9 + 321873492*x^8 - 408815960*x^7 - 263064528*x^6 + 88908581*x^5 + 40531608*x^4 - 2051852*x^3 - 1066000*x^2 + 23968*x + 4768, 1)
 

Normalized defining polynomial

\( x^{25} - x^{24} - 120 x^{23} + 87 x^{22} + 5789 x^{21} - 2591 x^{20} - 146272 x^{19} + 26182 x^{18} + 2128729 x^{17} + 143339 x^{16} - 18517488 x^{15} - 5136393 x^{14} + 96923558 x^{13} + 42784540 x^{12} - 299088119 x^{11} - 167117297 x^{10} + 510501238 x^{9} + 321873492 x^{8} - 408815960 x^{7} - 263064528 x^{6} + 88908581 x^{5} + 40531608 x^{4} - 2051852 x^{3} - 1066000 x^{2} + 23968 x + 4768 \)

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:  \(3909933000788329339715778837921617322655826705601954756001=251^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $201.23$
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:  $251$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(251\)
Dirichlet character group:    $\lbrace$$\chi_{251}(64,·)$, $\chi_{251}(1,·)$, $\chi_{251}(4,·)$, $\chi_{251}(5,·)$, $\chi_{251}(243,·)$, $\chi_{251}(201,·)$, $\chi_{251}(204,·)$, $\chi_{251}(16,·)$, $\chi_{251}(211,·)$, $\chi_{251}(20,·)$, $\chi_{251}(149,·)$, $\chi_{251}(25,·)$, $\chi_{251}(91,·)$, $\chi_{251}(94,·)$, $\chi_{251}(69,·)$, $\chi_{251}(80,·)$, $\chi_{251}(219,·)$, $\chi_{251}(100,·)$, $\chi_{251}(241,·)$, $\chi_{251}(113,·)$, $\chi_{251}(51,·)$, $\chi_{251}(249,·)$, $\chi_{251}(123,·)$, $\chi_{251}(125,·)$, $\chi_{251}(63,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{2} a^{21} - \frac{1}{2} a^{20} - \frac{1}{2} a^{18} - \frac{1}{2} a^{17} - \frac{1}{2} a^{16} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a$, $\frac{1}{4} a^{22} - \frac{1}{4} a^{21} - \frac{1}{4} a^{19} + \frac{1}{4} a^{18} + \frac{1}{4} a^{17} - \frac{1}{2} a^{15} + \frac{1}{4} a^{14} - \frac{1}{4} a^{13} - \frac{1}{4} a^{11} - \frac{1}{2} a^{10} + \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{2} a^{6} + \frac{1}{4} a^{2}$, $\frac{1}{121208485432} a^{23} - \frac{10370875917}{121208485432} a^{22} + \frac{489511409}{15151060679} a^{21} - \frac{26994025605}{121208485432} a^{20} + \frac{39818211969}{121208485432} a^{19} + \frac{45796356201}{121208485432} a^{18} - \frac{2196425189}{15151060679} a^{17} + \frac{4487849885}{60604242716} a^{16} - \frac{49422091839}{121208485432} a^{15} - \frac{18634361177}{121208485432} a^{14} + \frac{4871564544}{15151060679} a^{13} - \frac{11884573277}{121208485432} a^{12} - \frac{19779884391}{60604242716} a^{11} + \frac{857523633}{15151060679} a^{10} - \frac{17086787575}{121208485432} a^{9} + \frac{48549652995}{121208485432} a^{8} - \frac{20853987581}{60604242716} a^{7} + \frac{5782733533}{15151060679} a^{6} + \frac{5972622785}{15151060679} a^{5} + \frac{1011827880}{15151060679} a^{4} + \frac{58225102741}{121208485432} a^{3} + \frac{2058799183}{30302121358} a^{2} - \frac{1589027898}{15151060679} a + \frac{41870118}{101684971}$, $\frac{1}{849002674435058142785698883911957044217974814738987868459101392738951598972463504} a^{24} + \frac{2685198823055260062050152367199729445372512150806488723434098840590093}{849002674435058142785698883911957044217974814738987868459101392738951598972463504} a^{23} - \frac{32788764780947729013378241182072725306138504901295987754016405439039477086483831}{424501337217529071392849441955978522108987407369493934229550696369475799486231752} a^{22} + \frac{102300888351397044077781688319438637616676994371912408083510554719472841184633927}{849002674435058142785698883911957044217974814738987868459101392738951598972463504} a^{21} - \frac{413063950512667622278795549541340153353931388304043191906850033509746960255735241}{849002674435058142785698883911957044217974814738987868459101392738951598972463504} a^{20} + \frac{358221602841819864006926120335815364969913318103352281583920559057383988312523647}{849002674435058142785698883911957044217974814738987868459101392738951598972463504} a^{19} + \frac{70358159661979110011195238440405728175265934376917342309514508853209390851043815}{424501337217529071392849441955978522108987407369493934229550696369475799486231752} a^{18} + \frac{187079513878482780024113919017190660905587784105568448467309576734096308102104107}{424501337217529071392849441955978522108987407369493934229550696369475799486231752} a^{17} + \frac{111506319529548410868561695896192836433612658787474203133726568151765968707331253}{849002674435058142785698883911957044217974814738987868459101392738951598972463504} a^{16} + \frac{297380614821587939529318279148149308540562943674021696871019045162491660131266393}{849002674435058142785698883911957044217974814738987868459101392738951598972463504} a^{15} - \frac{41371211191301174772656963478924923330562811886391300729125838203926325148110215}{424501337217529071392849441955978522108987407369493934229550696369475799486231752} a^{14} + \frac{204071284032248534108024950821802184597415781035751302502644242754887843617877223}{849002674435058142785698883911957044217974814738987868459101392738951598972463504} a^{13} + \frac{22374288605875539377496183515870692891768134584739048541572959142180472903531132}{53062667152191133924106180244497315263623425921186741778693837046184474935778969} a^{12} + \frac{5595753074156370717746522390645600909511235169669887837449240420796868467986375}{106125334304382267848212360488994630527246851842373483557387674092368949871557938} a^{11} - \frac{259855452154240069900897767457151662850628994784152609252295392362512605730170951}{849002674435058142785698883911957044217974814738987868459101392738951598972463504} a^{10} + \frac{117726673448820437905256465970069034109692577203996165015975340564522496813139821}{849002674435058142785698883911957044217974814738987868459101392738951598972463504} a^{9} - \frac{26202416958836623787892460848486475753817753363900214584647226254986763619967206}{53062667152191133924106180244497315263623425921186741778693837046184474935778969} a^{8} + \frac{7807110961369691827443853158155012358513125757757766240953213613228201114925299}{106125334304382267848212360488994630527246851842373483557387674092368949871557938} a^{7} - \frac{25229915108745756386193459900503439689212615918073000031990138649725055313576051}{106125334304382267848212360488994630527246851842373483557387674092368949871557938} a^{6} - \frac{18889724672769491540309475177424993465521993155392203391623276280515468625923008}{53062667152191133924106180244497315263623425921186741778693837046184474935778969} a^{5} + \frac{339430282821266298223830338000565504034003354500826910414805612916111831657852357}{849002674435058142785698883911957044217974814738987868459101392738951598972463504} a^{4} - \frac{12022346704067651247459125316526226731888458697166591912505462114294379333659465}{424501337217529071392849441955978522108987407369493934229550696369475799486231752} a^{3} - \frac{64181542668395966797422662261457054564759100079720298587100961684286961086521145}{212250668608764535696424720977989261054493703684746967114775348184737899743115876} a^{2} + \frac{6177793909777087548709504821757927176999768366300300796273144614768525264884212}{53062667152191133924106180244497315263623425921186741778693837046184474935778969} a + \frac{114524806900702941789216501097701727765922110711410924916968805285704613454717}{356125282900611637074538122446290706467271314907293568984522396283117281448181}$

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:  \( 795868312785508000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{25}$ (as 25T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 25
The 25 conjugacy class representatives for $C_{25}$
Character table for $C_{25}$ is not computed

Intermediate fields

5.5.3969126001.1

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 ${\href{/LocalNumberField/2.5.0.1}{5} }^{5}$ $25$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{5}$ $25$ $25$ $25$ $25$ $25$ $25$ $25$ $25$ $25$ $25$ $25$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{5}$ $25$ $25$

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
251Data not computed