Properties

Label 25.1.145...641.1
Degree $25$
Signature $[1, 12]$
Discriminant $1.459\times 10^{32}$
Root discriminant $19.34$
Ramified prime $479$
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 - 3*x^24 + 9*x^23 - 22*x^22 + 41*x^21 - 60*x^20 + 66*x^19 - 47*x^18 + 6*x^17 + 48*x^16 - 82*x^15 + 76*x^14 + 11*x^13 - 138*x^12 + 280*x^11 - 336*x^10 + 317*x^9 - 205*x^8 + 144*x^7 - 109*x^6 + 126*x^5 - 104*x^4 + 76*x^3 - 23*x^2 + 10*x + 1)
 
gp: K = bnfinit(x^25 - 3*x^24 + 9*x^23 - 22*x^22 + 41*x^21 - 60*x^20 + 66*x^19 - 47*x^18 + 6*x^17 + 48*x^16 - 82*x^15 + 76*x^14 + 11*x^13 - 138*x^12 + 280*x^11 - 336*x^10 + 317*x^9 - 205*x^8 + 144*x^7 - 109*x^6 + 126*x^5 - 104*x^4 + 76*x^3 - 23*x^2 + 10*x + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 10, -23, 76, -104, 126, -109, 144, -205, 317, -336, 280, -138, 11, 76, -82, 48, 6, -47, 66, -60, 41, -22, 9, -3, 1]);
 

\( x^{25} - 3 x^{24} + 9 x^{23} - 22 x^{22} + 41 x^{21} - 60 x^{20} + 66 x^{19} - 47 x^{18} + 6 x^{17} + 48 x^{16} - 82 x^{15} + 76 x^{14} + 11 x^{13} - 138 x^{12} + 280 x^{11} - 336 x^{10} + 317 x^{9} - 205 x^{8} + 144 x^{7} - 109 x^{6} + 126 x^{5} - 104 x^{4} + 76 x^{3} - 23 x^{2} + 10 x + 1 \)

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:  \(145890213878661931676924574560641\)\(\medspace = 479^{12}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $19.34$
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:  $479$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $\frac{1}{403} a^{22} + \frac{183}{403} a^{21} + \frac{8}{31} a^{20} - \frac{119}{403} a^{19} - \frac{129}{403} a^{18} - \frac{121}{403} a^{17} - \frac{120}{403} a^{16} + \frac{8}{31} a^{15} + \frac{82}{403} a^{14} - \frac{139}{403} a^{13} - \frac{144}{403} a^{12} - \frac{69}{403} a^{11} + \frac{127}{403} a^{10} + \frac{25}{403} a^{9} - \frac{38}{403} a^{8} + \frac{154}{403} a^{7} + \frac{198}{403} a^{6} - \frac{4}{403} a^{5} + \frac{166}{403} a^{4} + \frac{6}{13} a^{3} + \frac{140}{403} a^{2} - \frac{20}{403} a - \frac{171}{403}$, $\frac{1}{280891} a^{23} - \frac{19}{16523} a^{22} + \frac{111021}{280891} a^{21} - \frac{54355}{280891} a^{20} - \frac{60815}{280891} a^{19} + \frac{90945}{280891} a^{18} - \frac{101706}{280891} a^{17} + \frac{139409}{280891} a^{16} + \frac{86896}{280891} a^{15} - \frac{99663}{280891} a^{14} - \frac{133728}{280891} a^{13} + \frac{72392}{280891} a^{12} - \frac{3320}{16523} a^{11} + \frac{68350}{280891} a^{10} - \frac{4727}{21607} a^{9} - \frac{83383}{280891} a^{8} + \frac{17382}{280891} a^{7} + \frac{4360}{9061} a^{6} - \frac{131203}{280891} a^{5} + \frac{52807}{280891} a^{4} + \frac{80523}{280891} a^{3} - \frac{47083}{280891} a^{2} - \frac{1169}{16523} a - \frac{50091}{280891}$, $\frac{1}{4572123216261133} a^{24} - \frac{6541108081}{4572123216261133} a^{23} + \frac{1205898133468}{4572123216261133} a^{22} + \frac{1738422709068255}{4572123216261133} a^{21} - \frac{2507231126183}{111515200396613} a^{20} - \frac{361345718106582}{4572123216261133} a^{19} - \frac{864047747781664}{4572123216261133} a^{18} + \frac{1584738786783467}{4572123216261133} a^{17} + \frac{628616268405611}{4572123216261133} a^{16} + \frac{168813110239627}{4572123216261133} a^{15} - \frac{1406025514253886}{4572123216261133} a^{14} + \frac{1544890378457310}{4572123216261133} a^{13} + \frac{851364700477942}{4572123216261133} a^{12} + \frac{695491991136685}{4572123216261133} a^{11} - \frac{2264002252329411}{4572123216261133} a^{10} + \frac{1318049170905160}{4572123216261133} a^{9} - \frac{1204982718753738}{4572123216261133} a^{8} + \frac{855305453036068}{4572123216261133} a^{7} - \frac{2022079813774835}{4572123216261133} a^{6} - \frac{1739410028250342}{4572123216261133} a^{5} + \frac{1948773131806504}{4572123216261133} a^{4} - \frac{2124698342039973}{4572123216261133} a^{3} - \frac{142489639201522}{351701785866241} a^{2} - \frac{80694613063309}{4572123216261133} a - \frac{2161938094563281}{4572123216261133}$

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:  \( 593552.1381003528 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{1}\cdot(2\pi)^{12}\cdot 593552.1381003528 \cdot 1}{2\sqrt{145890213878661931676924574560641}}\approx 0.186039093014903$ (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

5.1.229441.1

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 $25$ $25$ $25$ $25$ $25$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ $25$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
479Data not computed

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.479.2t1.a.a$1$ $ 479 $ \(\Q(\sqrt{-479}) \) $C_2$ (as 2T1) $1$ $-1$
* 2.479.5t2.a.a$2$ $ 479 $ 5.1.229441.1 $D_{5}$ (as 5T2) $1$ $0$
* 2.479.5t2.a.b$2$ $ 479 $ 5.1.229441.1 $D_{5}$ (as 5T2) $1$ $0$
* 2.479.25t4.a.f$2$ $ 479 $ 25.1.145890213878661931676924574560641.1 $D_{25}$ (as 25T4) $1$ $0$
* 2.479.25t4.a.d$2$ $ 479 $ 25.1.145890213878661931676924574560641.1 $D_{25}$ (as 25T4) $1$ $0$
* 2.479.25t4.a.h$2$ $ 479 $ 25.1.145890213878661931676924574560641.1 $D_{25}$ (as 25T4) $1$ $0$
* 2.479.25t4.a.c$2$ $ 479 $ 25.1.145890213878661931676924574560641.1 $D_{25}$ (as 25T4) $1$ $0$
* 2.479.25t4.a.i$2$ $ 479 $ 25.1.145890213878661931676924574560641.1 $D_{25}$ (as 25T4) $1$ $0$
* 2.479.25t4.a.g$2$ $ 479 $ 25.1.145890213878661931676924574560641.1 $D_{25}$ (as 25T4) $1$ $0$
* 2.479.25t4.a.a$2$ $ 479 $ 25.1.145890213878661931676924574560641.1 $D_{25}$ (as 25T4) $1$ $0$
* 2.479.25t4.a.j$2$ $ 479 $ 25.1.145890213878661931676924574560641.1 $D_{25}$ (as 25T4) $1$ $0$
* 2.479.25t4.a.e$2$ $ 479 $ 25.1.145890213878661931676924574560641.1 $D_{25}$ (as 25T4) $1$ $0$
* 2.479.25t4.a.b$2$ $ 479 $ 25.1.145890213878661931676924574560641.1 $D_{25}$ (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 *.