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Properties

Label 25.25.1974452703...3601.1
Degree $25$
Signature $[25, 0]$
Discriminant $151^{24}$
Root discriminant $123.54$
Ramified prime $151$
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![-111103, -190198, 5029783, -14814116, 5737119, 38006340, -55689151, -9872226, 75903945, -42180110, -30337248, 39567816, -4263646, -12138942, 5134575, 1274267, -1219472, 59219, 131523, -23789, -6935, 1991, 161, -72, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^25 - x^24 - 72*x^23 + 161*x^22 + 1991*x^21 - 6935*x^20 - 23789*x^19 + 131523*x^18 + 59219*x^17 - 1219472*x^16 + 1274267*x^15 + 5134575*x^14 - 12138942*x^13 - 4263646*x^12 + 39567816*x^11 - 30337248*x^10 - 42180110*x^9 + 75903945*x^8 - 9872226*x^7 - 55689151*x^6 + 38006340*x^5 + 5737119*x^4 - 14814116*x^3 + 5029783*x^2 - 190198*x - 111103)
 
gp: K = bnfinit(x^25 - x^24 - 72*x^23 + 161*x^22 + 1991*x^21 - 6935*x^20 - 23789*x^19 + 131523*x^18 + 59219*x^17 - 1219472*x^16 + 1274267*x^15 + 5134575*x^14 - 12138942*x^13 - 4263646*x^12 + 39567816*x^11 - 30337248*x^10 - 42180110*x^9 + 75903945*x^8 - 9872226*x^7 - 55689151*x^6 + 38006340*x^5 + 5737119*x^4 - 14814116*x^3 + 5029783*x^2 - 190198*x - 111103, 1)
 

Normalized defining polynomial

\( x^{25} - x^{24} - 72 x^{23} + 161 x^{22} + 1991 x^{21} - 6935 x^{20} - 23789 x^{19} + 131523 x^{18} + 59219 x^{17} - 1219472 x^{16} + 1274267 x^{15} + 5134575 x^{14} - 12138942 x^{13} - 4263646 x^{12} + 39567816 x^{11} - 30337248 x^{10} - 42180110 x^{9} + 75903945 x^{8} - 9872226 x^{7} - 55689151 x^{6} + 38006340 x^{5} + 5737119 x^{4} - 14814116 x^{3} + 5029783 x^{2} - 190198 x - 111103 \)

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:  \(19744527036368077698033828496963106435582783749713601=151^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $123.54$
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:  $151$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(151\)
Dirichlet character group:    $\lbrace$$\chi_{151}(64,·)$, $\chi_{151}(1,·)$, $\chi_{151}(68,·)$, $\chi_{151}(8,·)$, $\chi_{151}(9,·)$, $\chi_{151}(78,·)$, $\chi_{151}(81,·)$, $\chi_{151}(19,·)$, $\chi_{151}(148,·)$, $\chi_{151}(86,·)$, $\chi_{151}(20,·)$, $\chi_{151}(91,·)$, $\chi_{151}(29,·)$, $\chi_{151}(94,·)$, $\chi_{151}(98,·)$, $\chi_{151}(123,·)$, $\chi_{151}(44,·)$, $\chi_{151}(110,·)$, $\chi_{151}(72,·)$, $\chi_{151}(50,·)$, $\chi_{151}(84,·)$, $\chi_{151}(59,·)$, $\chi_{151}(124,·)$, $\chi_{151}(125,·)$, $\chi_{151}(127,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{2} a^{20} - \frac{1}{2} a^{19} - \frac{1}{2} a^{18} - \frac{1}{2} a^{17} - \frac{1}{2} a^{15} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{21} - \frac{1}{2} a^{17} - \frac{1}{2} a^{16} - \frac{1}{2} a^{15} - \frac{1}{2} a^{13} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{22} - \frac{1}{2} a^{18} - \frac{1}{2} a^{17} - \frac{1}{2} a^{16} - \frac{1}{2} a^{14} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{9902} a^{23} + \frac{825}{4951} a^{22} + \frac{743}{9902} a^{21} - \frac{899}{9902} a^{20} + \frac{2090}{4951} a^{19} + \frac{608}{4951} a^{18} + \frac{367}{9902} a^{17} - \frac{249}{9902} a^{16} - \frac{983}{9902} a^{15} + \frac{560}{4951} a^{14} + \frac{2205}{9902} a^{13} + \frac{4473}{9902} a^{12} - \frac{707}{9902} a^{11} - \frac{1027}{4951} a^{10} - \frac{1791}{9902} a^{9} - \frac{2493}{9902} a^{8} - \frac{3721}{9902} a^{7} + \frac{1748}{4951} a^{6} - \frac{2441}{4951} a^{5} + \frac{3485}{9902} a^{4} - \frac{1675}{9902} a^{3} + \frac{1510}{4951} a^{2} - \frac{3233}{9902} a - \frac{82}{4951}$, $\frac{1}{31335750610491467928713004214203381748701982} a^{24} - \frac{1414994768870322738520550467726262122519}{31335750610491467928713004214203381748701982} a^{23} - \frac{785613088551598966794759090262989592587209}{15667875305245733964356502107101690874350991} a^{22} + \frac{87588958167476472392364034901729789573636}{15667875305245733964356502107101690874350991} a^{21} + \frac{2874401340896817677891309660619515750037936}{15667875305245733964356502107101690874350991} a^{20} + \frac{14296069071368327650884207111820747447787}{116489779221157873340940536112280229549078} a^{19} + \frac{5631024277406501898970251623381012160177837}{31335750610491467928713004214203381748701982} a^{18} + \frac{3252684240727596451185476026670866345422698}{15667875305245733964356502107101690874350991} a^{17} - \frac{25126237428539574062321129047811808048865}{74431711663875220733285045639437961398342} a^{16} + \frac{4436809223869170542584810194762717151166716}{15667875305245733964356502107101690874350991} a^{15} - \frac{3383805693667320294240567335796768070162851}{15667875305245733964356502107101690874350991} a^{14} + \frac{5292426683763047193746837817856059216522838}{15667875305245733964356502107101690874350991} a^{13} - \frac{7681541447709460036886735228258800903552069}{31335750610491467928713004214203381748701982} a^{12} - \frac{1858518415908326861360889365680059621480543}{15667875305245733964356502107101690874350991} a^{11} + \frac{7448904577138316857874780339508251889987707}{15667875305245733964356502107101690874350991} a^{10} - \frac{4985878330478104578480032727822462736615612}{15667875305245733964356502107101690874350991} a^{9} - \frac{6397954204946101676619943049473537305687561}{31335750610491467928713004214203381748701982} a^{8} - \frac{6297798984237911890879487471942920049362202}{15667875305245733964356502107101690874350991} a^{7} - \frac{14464693768642534382949826906047020721048411}{31335750610491467928713004214203381748701982} a^{6} - \frac{8566132481832902859501959877299783177643705}{31335750610491467928713004214203381748701982} a^{5} - \frac{1939689839997217579875230936265232177504804}{15667875305245733964356502107101690874350991} a^{4} + \frac{5726408672171585031297737403653038926413426}{15667875305245733964356502107101690874350991} a^{3} - \frac{4769726543620585356127431422923609073514811}{15667875305245733964356502107101690874350991} a^{2} + \frac{4795645749495432919590091715900465186600335}{31335750610491467928713004214203381748701982} a + \frac{375494654381132835782370572989183070921865}{31335750610491467928713004214203381748701982}$

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:  \( 1066553081453288300 \) (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

5.5.519885601.1

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 ${\href{/LocalNumberField/2.5.0.1}{5} }^{5}$ $25$ $25$ $25$ $25$ $25$ $25$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{5}$ $25$ $25$ $25$ $25$ $25$ $25$ $25$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{5}$

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
151Data not computed