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Properties

Label 25.1.213...801.1
Degree $25$
Signature $[1, 12]$
Discriminant $2.134\times 10^{33}$
Root discriminant $21.54$
Ramified prime $599$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $D_{25}$ (as 25T4)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^25 - 2*x^24 - x^23 + 8*x^22 + 6*x^21 - 17*x^20 - 5*x^19 + 32*x^18 - 13*x^17 - 36*x^16 + 22*x^15 + 58*x^14 - 6*x^13 - 71*x^12 + 35*x^10 - 24*x^9 - 66*x^8 + 28*x^7 + 63*x^6 + 92*x^5 - 82*x^4 - 3*x^3 + 25*x^2 + 16*x - 1)
 
gp: K = bnfinit(x^25 - 2*x^24 - x^23 + 8*x^22 + 6*x^21 - 17*x^20 - 5*x^19 + 32*x^18 - 13*x^17 - 36*x^16 + 22*x^15 + 58*x^14 - 6*x^13 - 71*x^12 + 35*x^10 - 24*x^9 - 66*x^8 + 28*x^7 + 63*x^6 + 92*x^5 - 82*x^4 - 3*x^3 + 25*x^2 + 16*x - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 16, 25, -3, -82, 92, 63, 28, -66, -24, 35, 0, -71, -6, 58, 22, -36, -13, 32, -5, -17, 6, 8, -1, -2, 1]);
 

\( x^{25} - 2 x^{24} - x^{23} + 8 x^{22} + 6 x^{21} - 17 x^{20} - 5 x^{19} + 32 x^{18} - 13 x^{17} - 36 x^{16} + 22 x^{15} + 58 x^{14} - 6 x^{13} - 71 x^{12} + 35 x^{10} - 24 x^{9} - 66 x^{8} + 28 x^{7} + 63 x^{6} + 92 x^{5} - 82 x^{4} - 3 x^{3} + 25 x^{2} + 16 x - 1 \)

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:  \(2133643557240451317422184503752801\)\(\medspace = 599^{12}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $21.54$
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:  $599$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{29} a^{21} - \frac{9}{29} a^{20} + \frac{4}{29} a^{19} + \frac{12}{29} a^{18} + \frac{7}{29} a^{17} + \frac{11}{29} a^{16} - \frac{4}{29} a^{15} + \frac{4}{29} a^{14} - \frac{7}{29} a^{13} - \frac{14}{29} a^{12} + \frac{3}{29} a^{11} + \frac{9}{29} a^{9} - \frac{11}{29} a^{8} + \frac{7}{29} a^{7} + \frac{13}{29} a^{6} - \frac{10}{29} a^{5} + \frac{11}{29} a^{4} - \frac{9}{29} a^{3} - \frac{14}{29} a^{2} + \frac{2}{29} a + \frac{7}{29}$, $\frac{1}{319} a^{22} + \frac{4}{319} a^{21} - \frac{84}{319} a^{20} + \frac{6}{319} a^{19} - \frac{98}{319} a^{18} - \frac{101}{319} a^{17} + \frac{10}{29} a^{16} + \frac{10}{319} a^{15} - \frac{42}{319} a^{14} + \frac{127}{319} a^{13} - \frac{63}{319} a^{12} - \frac{48}{319} a^{11} + \frac{125}{319} a^{10} + \frac{135}{319} a^{9} - \frac{49}{319} a^{8} + \frac{104}{319} a^{7} + \frac{130}{319} a^{6} - \frac{148}{319} a^{5} - \frac{1}{29} a^{4} + \frac{14}{319} a^{3} - \frac{151}{319} a^{2} + \frac{3}{29} a - \frac{141}{319}$, $\frac{1}{652993} a^{23} + \frac{9}{59363} a^{22} + \frac{483}{28391} a^{21} + \frac{28370}{652993} a^{20} + \frac{35111}{652993} a^{19} + \frac{190525}{652993} a^{18} - \frac{8550}{28391} a^{17} + \frac{24133}{652993} a^{16} + \frac{187974}{652993} a^{15} - \frac{70028}{652993} a^{14} + \frac{1706}{652993} a^{13} + \frac{56953}{652993} a^{12} - \frac{227196}{652993} a^{11} - \frac{30736}{652993} a^{10} - \frac{270474}{652993} a^{9} + \frac{302690}{652993} a^{8} + \frac{2107}{59363} a^{7} + \frac{250066}{652993} a^{6} + \frac{10734}{28391} a^{5} - \frac{119743}{652993} a^{4} - \frac{34890}{652993} a^{3} - \frac{218010}{652993} a^{2} - \frac{84159}{652993} a + \frac{302184}{652993}$, $\frac{1}{41149686193745383793} a^{24} - \frac{549603224727}{5878526599106483399} a^{23} + \frac{4425522233192850}{41149686193745383793} a^{22} - \frac{692045513973704185}{41149686193745383793} a^{21} + \frac{16504015697361346683}{41149686193745383793} a^{20} - \frac{147238717079326550}{462356024648824537} a^{19} + \frac{4475479247711628259}{41149686193745383793} a^{18} - \frac{1847885909167483728}{5878526599106483399} a^{17} - \frac{242156139214007128}{3740880563067762163} a^{16} + \frac{1503138932627339372}{41149686193745383793} a^{15} + \frac{8779820669501186952}{41149686193745383793} a^{14} + \frac{18371859235671824394}{41149686193745383793} a^{13} + \frac{10761005510957681395}{41149686193745383793} a^{12} - \frac{5903764714366590829}{41149686193745383793} a^{11} - \frac{1069324996510313362}{3740880563067762163} a^{10} + \frac{7512852319340889793}{41149686193745383793} a^{9} + \frac{19810768602948602918}{41149686193745383793} a^{8} - \frac{6851766220208549499}{41149686193745383793} a^{7} - \frac{10000792882886463607}{41149686193745383793} a^{6} + \frac{16687387147620332623}{41149686193745383793} a^{5} - \frac{17613115956058858380}{41149686193745383793} a^{4} + \frac{220162729765273469}{1418954696336047717} a^{3} - \frac{408297830993109944}{41149686193745383793} a^{2} + \frac{16805752258218084792}{41149686193745383793} a - \frac{12590489922324248259}{41149686193745383793}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( 3027380.0731932656 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{1}\cdot(2\pi)^{12}\cdot 3027380.0731932656 \cdot 1}{2\sqrt{2133643557240451317422184503752801}}\approx 0.248121520542001$ (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

5.1.358801.1

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 $25$ $25$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ $25$ $25$ $25$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ $25$ $25$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{5}$ $25$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
599Data not computed

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.599.2t1.a.a$1$ $ 599 $ \(\Q(\sqrt{-599}) \) $C_2$ (as 2T1) $1$ $-1$
* 2.599.5t2.a.a$2$ $ 599 $ 5.1.358801.1 $D_{5}$ (as 5T2) $1$ $0$
* 2.599.5t2.a.b$2$ $ 599 $ 5.1.358801.1 $D_{5}$ (as 5T2) $1$ $0$
* 2.599.25t4.a.e$2$ $ 599 $ 25.1.2133643557240451317422184503752801.1 $D_{25}$ (as 25T4) $1$ $0$
* 2.599.25t4.a.j$2$ $ 599 $ 25.1.2133643557240451317422184503752801.1 $D_{25}$ (as 25T4) $1$ $0$
* 2.599.25t4.a.c$2$ $ 599 $ 25.1.2133643557240451317422184503752801.1 $D_{25}$ (as 25T4) $1$ $0$
* 2.599.25t4.a.i$2$ $ 599 $ 25.1.2133643557240451317422184503752801.1 $D_{25}$ (as 25T4) $1$ $0$
* 2.599.25t4.a.f$2$ $ 599 $ 25.1.2133643557240451317422184503752801.1 $D_{25}$ (as 25T4) $1$ $0$
* 2.599.25t4.a.a$2$ $ 599 $ 25.1.2133643557240451317422184503752801.1 $D_{25}$ (as 25T4) $1$ $0$
* 2.599.25t4.a.b$2$ $ 599 $ 25.1.2133643557240451317422184503752801.1 $D_{25}$ (as 25T4) $1$ $0$
* 2.599.25t4.a.h$2$ $ 599 $ 25.1.2133643557240451317422184503752801.1 $D_{25}$ (as 25T4) $1$ $0$
* 2.599.25t4.a.g$2$ $ 599 $ 25.1.2133643557240451317422184503752801.1 $D_{25}$ (as 25T4) $1$ $0$
* 2.599.25t4.a.d$2$ $ 599 $ 25.1.2133643557240451317422184503752801.1 $D_{25}$ (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 *.