Properties

Label 25.1.114...401.1
Degree $25$
Signature $[1, 12]$
Discriminant $1.149\times 10^{39}$
Root discriminant $36.51$
Ramified primes $7, 257$
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 - 4*x^24 + x^23 + 2*x^22 + 5*x^21 + 38*x^20 + 24*x^19 - 79*x^18 - 45*x^17 - 175*x^16 - 246*x^15 + 78*x^14 + 160*x^13 + 28*x^12 + 1032*x^11 + 953*x^10 + 967*x^9 + 1690*x^8 + 1081*x^7 - 1035*x^6 - 715*x^5 - 2390*x^4 - 2966*x^3 - 1925*x^2 - 720*x - 1003)
 
gp: K = bnfinit(x^25 - 4*x^24 + x^23 + 2*x^22 + 5*x^21 + 38*x^20 + 24*x^19 - 79*x^18 - 45*x^17 - 175*x^16 - 246*x^15 + 78*x^14 + 160*x^13 + 28*x^12 + 1032*x^11 + 953*x^10 + 967*x^9 + 1690*x^8 + 1081*x^7 - 1035*x^6 - 715*x^5 - 2390*x^4 - 2966*x^3 - 1925*x^2 - 720*x - 1003, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1003, -720, -1925, -2966, -2390, -715, -1035, 1081, 1690, 967, 953, 1032, 28, 160, 78, -246, -175, -45, -79, 24, 38, 5, 2, 1, -4, 1]);
 

\( x^{25} - 4 x^{24} + x^{23} + 2 x^{22} + 5 x^{21} + 38 x^{20} + 24 x^{19} - 79 x^{18} - 45 x^{17} - 175 x^{16} - 246 x^{15} + 78 x^{14} + 160 x^{13} + 28 x^{12} + 1032 x^{11} + 953 x^{10} + 967 x^{9} + 1690 x^{8} + 1081 x^{7} - 1035 x^{6} - 715 x^{5} - 2390 x^{4} - 2966 x^{3} - 1925 x^{2} - 720 x - 1003 \)

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:  \(1149142693715820345877000392469173818401\)\(\medspace = 7^{12}\cdot 257^{12}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $36.51$
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:  $7, 257$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{7} a^{11} - \frac{2}{7} a^{10} - \frac{3}{7} a^{9} - \frac{1}{7} a^{8} + \frac{2}{7} a^{7} + \frac{3}{7} a^{6} - \frac{1}{7} a^{5} + \frac{2}{7} a^{4} + \frac{3}{7} a^{3} + \frac{1}{7} a^{2} - \frac{2}{7} a - \frac{3}{7}$, $\frac{1}{7} a^{12} - \frac{2}{7} a^{6} + \frac{1}{7}$, $\frac{1}{7} a^{13} - \frac{2}{7} a^{7} + \frac{1}{7} a$, $\frac{1}{7} a^{14} - \frac{2}{7} a^{8} + \frac{1}{7} a^{2}$, $\frac{1}{7} a^{15} - \frac{2}{7} a^{9} + \frac{1}{7} a^{3}$, $\frac{1}{7} a^{16} - \frac{2}{7} a^{10} + \frac{1}{7} a^{4}$, $\frac{1}{49} a^{17} - \frac{3}{49} a^{16} - \frac{3}{49} a^{15} + \frac{2}{49} a^{14} + \frac{2}{49} a^{13} + \frac{1}{49} a^{12} - \frac{2}{49} a^{11} + \frac{6}{49} a^{10} - \frac{15}{49} a^{9} + \frac{3}{49} a^{8} + \frac{10}{49} a^{7} - \frac{9}{49} a^{6} - \frac{20}{49} a^{5} + \frac{4}{49} a^{4} - \frac{17}{49} a^{3} + \frac{23}{49} a^{2} - \frac{5}{49} a + \frac{22}{49}$, $\frac{1}{49} a^{18} + \frac{2}{49} a^{16} + \frac{1}{49} a^{14} + \frac{1}{49} a^{12} + \frac{24}{49} a^{10} - \frac{1}{7} a^{9} - \frac{16}{49} a^{8} - \frac{2}{7} a^{7} + \frac{2}{49} a^{6} - \frac{1}{7} a^{5} + \frac{9}{49} a^{4} - \frac{3}{7} a^{3} + \frac{8}{49} a^{2} + \frac{17}{49}$, $\frac{1}{637} a^{19} - \frac{2}{637} a^{18} - \frac{3}{637} a^{17} + \frac{25}{637} a^{16} - \frac{33}{637} a^{15} + \frac{30}{637} a^{14} + \frac{2}{49} a^{13} + \frac{4}{91} a^{12} + \frac{34}{637} a^{11} - \frac{15}{637} a^{10} + \frac{220}{637} a^{9} - \frac{81}{637} a^{8} - \frac{22}{49} a^{7} - \frac{134}{637} a^{6} + \frac{74}{637} a^{5} - \frac{192}{637} a^{4} + \frac{135}{637} a^{3} + \frac{303}{637} a^{2} - \frac{38}{91} a - \frac{158}{637}$, $\frac{1}{637} a^{20} + \frac{6}{637} a^{18} + \frac{6}{637} a^{17} - \frac{9}{637} a^{16} + \frac{3}{637} a^{15} - \frac{18}{637} a^{14} - \frac{37}{637} a^{13} - \frac{1}{637} a^{12} - \frac{12}{637} a^{11} + \frac{151}{637} a^{10} + \frac{99}{637} a^{9} + \frac{215}{637} a^{8} + \frac{256}{637} a^{7} - \frac{142}{637} a^{6} + \frac{216}{637} a^{5} + \frac{180}{637} a^{4} + \frac{248}{637} a^{3} - \frac{37}{637} a^{2} + \frac{103}{637} a - \frac{199}{637}$, $\frac{1}{637} a^{21} + \frac{5}{637} a^{18} - \frac{4}{637} a^{17} - \frac{43}{637} a^{16} + \frac{37}{637} a^{15} + \frac{17}{637} a^{14} - \frac{1}{637} a^{13} - \frac{24}{637} a^{12} - \frac{27}{637} a^{11} + \frac{254}{637} a^{10} + \frac{2}{7} a^{9} - \frac{272}{637} a^{8} - \frac{12}{637} a^{7} + \frac{110}{637} a^{6} + \frac{87}{637} a^{5} + \frac{48}{637} a^{4} + \frac{102}{637} a^{3} + \frac{66}{637} a^{2} - \frac{267}{637} a - \frac{2}{91}$, $\frac{1}{4459} a^{22} + \frac{2}{4459} a^{21} + \frac{1}{4459} a^{20} - \frac{17}{4459} a^{18} + \frac{22}{4459} a^{17} + \frac{38}{4459} a^{16} + \frac{285}{4459} a^{15} + \frac{16}{637} a^{14} + \frac{93}{4459} a^{13} + \frac{23}{637} a^{12} - \frac{177}{4459} a^{11} + \frac{2112}{4459} a^{10} - \frac{233}{4459} a^{9} + \frac{1845}{4459} a^{8} - \frac{256}{4459} a^{7} - \frac{1986}{4459} a^{6} + \frac{1940}{4459} a^{5} + \frac{2105}{4459} a^{4} + \frac{324}{4459} a^{3} - \frac{1349}{4459} a^{2} + \frac{1626}{4459} a - \frac{2141}{4459}$, $\frac{1}{10464073529} a^{23} - \frac{729737}{10464073529} a^{22} - \frac{7075836}{10464073529} a^{21} - \frac{433219}{10464073529} a^{20} - \frac{915519}{10464073529} a^{19} + \frac{11805489}{10464073529} a^{18} + \frac{11556390}{1494867647} a^{17} + \frac{16858549}{337550759} a^{16} + \frac{220803491}{10464073529} a^{15} - \frac{101456206}{10464073529} a^{14} + \frac{514614983}{10464073529} a^{13} + \frac{230268631}{10464073529} a^{12} - \frac{404610146}{10464073529} a^{11} + \frac{2649485780}{10464073529} a^{10} - \frac{304820473}{615533737} a^{9} - \frac{484023374}{10464073529} a^{8} - \frac{11154938}{48221537} a^{7} - \frac{3495187197}{10464073529} a^{6} - \frac{4019645684}{10464073529} a^{5} + \frac{135206532}{804928733} a^{4} - \frac{4669651242}{10464073529} a^{3} + \frac{2468021643}{10464073529} a^{2} + \frac{402352137}{804928733} a + \frac{156605329}{615533737}$, $\frac{1}{5849417102711} a^{24} + \frac{121}{5849417102711} a^{23} + \frac{7813840}{95892083651} a^{22} - \frac{1946688777}{5849417102711} a^{21} + \frac{274447122}{5849417102711} a^{20} - \frac{2396008803}{5849417102711} a^{19} - \frac{1880871352}{835631014673} a^{18} + \frac{52653029652}{5849417102711} a^{17} - \frac{178451204376}{5849417102711} a^{16} - \frac{37743604855}{835631014673} a^{15} - \frac{52107438094}{835631014673} a^{14} + \frac{55250198222}{835631014673} a^{13} + \frac{144416524040}{5849417102711} a^{12} - \frac{312349803770}{5849417102711} a^{11} + \frac{1107771349008}{5849417102711} a^{10} + \frac{2771808646902}{5849417102711} a^{9} + \frac{1986841436761}{5849417102711} a^{8} + \frac{2199434628864}{5849417102711} a^{7} - \frac{163032060848}{835631014673} a^{6} + \frac{397755889534}{5849417102711} a^{5} - \frac{400475197217}{835631014673} a^{4} - \frac{2747940752896}{5849417102711} a^{3} - \frac{2557546410962}{5849417102711} a^{2} + \frac{2140695688518}{5849417102711} a + \frac{130832222127}{344083358983}$

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:  \( 11203233087.777065 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{1}\cdot(2\pi)^{12}\cdot 11203233087.777065 \cdot 1}{2\sqrt{1149142693715820345877000392469173818401}}\approx 1.25116564820286$ (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

5.1.3236401.1

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 $25$ $25$ $25$ R ${\href{/LocalNumberField/11.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ $25$ $25$ $25$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ $25$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

In the table, R denotes a ramified prime. 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
$7$$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
257Data not computed

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.1799.2t1.a.a$1$ $ 7 \cdot 257 $ \(\Q(\sqrt{-1799}) \) $C_2$ (as 2T1) $1$ $-1$
* 2.1799.5t2.a.a$2$ $ 7 \cdot 257 $ 5.1.3236401.1 $D_{5}$ (as 5T2) $1$ $0$
* 2.1799.5t2.a.b$2$ $ 7 \cdot 257 $ 5.1.3236401.1 $D_{5}$ (as 5T2) $1$ $0$
* 2.1799.25t4.a.h$2$ $ 7 \cdot 257 $ 25.1.1149142693715820345877000392469173818401.1 $D_{25}$ (as 25T4) $1$ $0$
* 2.1799.25t4.a.g$2$ $ 7 \cdot 257 $ 25.1.1149142693715820345877000392469173818401.1 $D_{25}$ (as 25T4) $1$ $0$
* 2.1799.25t4.a.b$2$ $ 7 \cdot 257 $ 25.1.1149142693715820345877000392469173818401.1 $D_{25}$ (as 25T4) $1$ $0$
* 2.1799.25t4.a.d$2$ $ 7 \cdot 257 $ 25.1.1149142693715820345877000392469173818401.1 $D_{25}$ (as 25T4) $1$ $0$
* 2.1799.25t4.a.j$2$ $ 7 \cdot 257 $ 25.1.1149142693715820345877000392469173818401.1 $D_{25}$ (as 25T4) $1$ $0$
* 2.1799.25t4.a.i$2$ $ 7 \cdot 257 $ 25.1.1149142693715820345877000392469173818401.1 $D_{25}$ (as 25T4) $1$ $0$
* 2.1799.25t4.a.f$2$ $ 7 \cdot 257 $ 25.1.1149142693715820345877000392469173818401.1 $D_{25}$ (as 25T4) $1$ $0$
* 2.1799.25t4.a.a$2$ $ 7 \cdot 257 $ 25.1.1149142693715820345877000392469173818401.1 $D_{25}$ (as 25T4) $1$ $0$
* 2.1799.25t4.a.c$2$ $ 7 \cdot 257 $ 25.1.1149142693715820345877000392469173818401.1 $D_{25}$ (as 25T4) $1$ $0$
* 2.1799.25t4.a.e$2$ $ 7 \cdot 257 $ 25.1.1149142693715820345877000392469173818401.1 $D_{25}$ (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 *.