Properties

Label 26.0.167...007.1
Degree $26$
Signature $[0, 13]$
Discriminant $-1.675\times 10^{42}$
Root discriminant $42.07$
Ramified primes $7, 401$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $D_{26}$ (as 26T3)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^26 - 8*x^25 + 22*x^24 - 43*x^23 + 115*x^22 - 151*x^21 + 220*x^20 - 577*x^19 + 509*x^18 - 1015*x^17 + 1547*x^16 - 884*x^15 + 2815*x^14 - 3027*x^13 + 2341*x^12 - 3581*x^11 + 6893*x^10 + 2193*x^9 + 12186*x^8 - 1094*x^7 + 906*x^6 - 2074*x^5 + 14041*x^4 + 14998*x^3 + 13037*x^2 + 4230*x + 1325)
 
gp: K = bnfinit(x^26 - 8*x^25 + 22*x^24 - 43*x^23 + 115*x^22 - 151*x^21 + 220*x^20 - 577*x^19 + 509*x^18 - 1015*x^17 + 1547*x^16 - 884*x^15 + 2815*x^14 - 3027*x^13 + 2341*x^12 - 3581*x^11 + 6893*x^10 + 2193*x^9 + 12186*x^8 - 1094*x^7 + 906*x^6 - 2074*x^5 + 14041*x^4 + 14998*x^3 + 13037*x^2 + 4230*x + 1325, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1325, 4230, 13037, 14998, 14041, -2074, 906, -1094, 12186, 2193, 6893, -3581, 2341, -3027, 2815, -884, 1547, -1015, 509, -577, 220, -151, 115, -43, 22, -8, 1]);
 

\( x^{26} - 8 x^{25} + 22 x^{24} - 43 x^{23} + 115 x^{22} - 151 x^{21} + 220 x^{20} - 577 x^{19} + 509 x^{18} - 1015 x^{17} + 1547 x^{16} - 884 x^{15} + 2815 x^{14} - 3027 x^{13} + 2341 x^{12} - 3581 x^{11} + 6893 x^{10} + 2193 x^{9} + 12186 x^{8} - 1094 x^{7} + 906 x^{6} - 2074 x^{5} + 14041 x^{4} + 14998 x^{3} + 13037 x^{2} + 4230 x + 1325 \)

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 

Invariants

Degree:  $26$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 13]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(-1674969840842867253564530506487708596884007\)\(\medspace = -\,7^{13}\cdot 401^{12}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $42.07$
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:  $7, 401$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $2$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{7} a^{12} - \frac{2}{7} a^{6} + \frac{1}{7}$, $\frac{1}{7} a^{13} - \frac{2}{7} a^{7} + \frac{1}{7} a$, $\frac{1}{35} a^{14} - \frac{2}{35} a^{13} - \frac{1}{35} a^{12} - \frac{1}{5} a^{10} - \frac{9}{35} a^{8} - \frac{2}{7} a^{7} + \frac{2}{35} a^{6} + \frac{2}{5} a^{4} - \frac{6}{35} a^{2} - \frac{2}{35} a - \frac{3}{7}$, $\frac{1}{35} a^{15} - \frac{2}{35} a^{12} - \frac{1}{5} a^{11} - \frac{2}{5} a^{10} - \frac{9}{35} a^{9} + \frac{1}{5} a^{8} + \frac{1}{5} a^{7} + \frac{4}{35} a^{6} + \frac{2}{5} a^{5} - \frac{1}{5} a^{4} - \frac{6}{35} a^{3} - \frac{2}{5} a^{2} - \frac{2}{5} a + \frac{1}{7}$, $\frac{1}{35} a^{16} - \frac{2}{35} a^{13} - \frac{2}{35} a^{12} - \frac{2}{5} a^{11} - \frac{9}{35} a^{10} + \frac{1}{5} a^{9} + \frac{1}{5} a^{8} + \frac{4}{35} a^{7} + \frac{4}{35} a^{6} - \frac{1}{5} a^{5} - \frac{6}{35} a^{4} - \frac{2}{5} a^{3} - \frac{2}{5} a^{2} + \frac{1}{7} a + \frac{1}{7}$, $\frac{1}{35} a^{17} - \frac{1}{35} a^{13} - \frac{1}{35} a^{12} - \frac{9}{35} a^{11} - \frac{1}{5} a^{10} + \frac{1}{5} a^{9} - \frac{2}{5} a^{8} + \frac{9}{35} a^{7} + \frac{2}{35} a^{6} - \frac{6}{35} a^{5} + \frac{2}{5} a^{4} - \frac{2}{5} a^{3} - \frac{1}{5} a^{2} + \frac{6}{35} a - \frac{3}{7}$, $\frac{1}{35} a^{18} + \frac{2}{35} a^{13} - \frac{1}{5} a^{11} - \frac{2}{5} a^{9} + \frac{17}{35} a^{7} + \frac{11}{35} a^{6} + \frac{2}{5} a^{5} - \frac{1}{5} a^{3} - \frac{12}{35} a - \frac{1}{7}$, $\frac{1}{245} a^{19} - \frac{3}{245} a^{18} - \frac{3}{245} a^{17} + \frac{2}{245} a^{16} + \frac{2}{245} a^{14} - \frac{1}{35} a^{13} + \frac{12}{245} a^{12} - \frac{3}{49} a^{11} + \frac{59}{245} a^{10} + \frac{2}{7} a^{9} - \frac{67}{245} a^{8} + \frac{116}{245} a^{7} + \frac{118}{245} a^{6} + \frac{67}{245} a^{5} + \frac{44}{245} a^{4} - \frac{1}{7} a^{3} + \frac{51}{245} a^{2} - \frac{47}{245} a - \frac{24}{49}$, $\frac{1}{245} a^{20} + \frac{2}{245} a^{18} - \frac{1}{245} a^{16} + \frac{2}{245} a^{15} - \frac{1}{245} a^{14} - \frac{9}{245} a^{13} - \frac{1}{35} a^{12} - \frac{1}{5} a^{11} + \frac{16}{245} a^{10} - \frac{53}{245} a^{9} + \frac{13}{245} a^{8} + \frac{74}{245} a^{7} - \frac{104}{245} a^{6} - \frac{6}{35} a^{5} - \frac{8}{245} a^{4} + \frac{93}{245} a^{3} - \frac{18}{49} a^{2} + \frac{33}{245} a - \frac{23}{49}$, $\frac{1}{245} a^{21} - \frac{1}{245} a^{18} - \frac{2}{245} a^{17} - \frac{2}{245} a^{16} - \frac{1}{245} a^{15} + \frac{1}{245} a^{14} + \frac{1}{35} a^{13} - \frac{2}{49} a^{12} - \frac{87}{245} a^{11} + \frac{5}{49} a^{10} - \frac{78}{245} a^{9} - \frac{13}{49} a^{8} + \frac{1}{35} a^{7} + \frac{9}{245} a^{6} + \frac{47}{245} a^{5} + \frac{103}{245} a^{4} - \frac{118}{245} a^{3} - \frac{104}{245} a^{2} + \frac{4}{35} a - \frac{1}{49}$, $\frac{1}{1715} a^{22} - \frac{1}{1715} a^{21} - \frac{3}{1715} a^{20} - \frac{3}{1715} a^{19} + \frac{4}{343} a^{18} - \frac{1}{1715} a^{17} - \frac{1}{245} a^{16} + \frac{3}{1715} a^{15} + \frac{12}{1715} a^{14} - \frac{32}{1715} a^{13} + \frac{5}{343} a^{12} - \frac{481}{1715} a^{11} - \frac{549}{1715} a^{10} - \frac{423}{1715} a^{9} + \frac{447}{1715} a^{8} - \frac{242}{1715} a^{7} - \frac{120}{343} a^{6} - \frac{204}{1715} a^{5} - \frac{292}{1715} a^{4} - \frac{433}{1715} a^{3} - \frac{673}{1715} a^{2} - \frac{94}{1715} a - \frac{127}{343}$, $\frac{1}{8575} a^{23} - \frac{1}{8575} a^{22} - \frac{17}{8575} a^{21} + \frac{11}{8575} a^{20} - \frac{3}{1715} a^{19} - \frac{2}{343} a^{18} - \frac{17}{1225} a^{17} + \frac{9}{1715} a^{16} - \frac{93}{8575} a^{15} + \frac{23}{1715} a^{14} - \frac{297}{8575} a^{13} - \frac{516}{8575} a^{12} + \frac{312}{8575} a^{11} + \frac{1992}{8575} a^{10} - \frac{66}{1715} a^{9} + \frac{2362}{8575} a^{8} + \frac{3824}{8575} a^{7} - \frac{3613}{8575} a^{6} + \frac{674}{8575} a^{5} - \frac{3086}{8575} a^{4} - \frac{2472}{8575} a^{3} - \frac{752}{8575} a^{2} - \frac{813}{1715} a + \frac{11}{49}$, $\frac{1}{317601822756575} a^{24} + \frac{691831426}{63520364551315} a^{23} + \frac{6074319142}{317601822756575} a^{22} + \frac{36213598852}{45371688965225} a^{21} - \frac{298727042784}{317601822756575} a^{20} - \frac{996412588}{870141980155} a^{19} - \frac{3853492857849}{317601822756575} a^{18} - \frac{2432779245714}{317601822756575} a^{17} + \frac{417181185496}{45371688965225} a^{16} + \frac{1342924470702}{317601822756575} a^{15} - \frac{887899463617}{317601822756575} a^{14} + \frac{11529964495762}{317601822756575} a^{13} + \frac{18420318267586}{317601822756575} a^{12} - \frac{90183453766211}{317601822756575} a^{11} + \frac{47965814675602}{317601822756575} a^{10} + \frac{149337225799637}{317601822756575} a^{9} + \frac{11116773618491}{317601822756575} a^{8} - \frac{53476221766474}{317601822756575} a^{7} + \frac{19841719747776}{317601822756575} a^{6} - \frac{129127472608262}{317601822756575} a^{5} + \frac{84434661836202}{317601822756575} a^{4} - \frac{31100940971314}{317601822756575} a^{3} - \frac{81760501521197}{317601822756575} a^{2} + \frac{2338473591047}{9074337793045} a - \frac{1261767400935}{12704072910263}$, $\frac{1}{3681322727571460825} a^{25} - \frac{102}{736264545514292165} a^{24} - \frac{4455789914238}{147252909102858433} a^{23} + \frac{948169752584611}{3681322727571460825} a^{22} + \frac{422994556187694}{736264545514292165} a^{21} + \frac{2804006951923043}{3681322727571460825} a^{20} + \frac{1947900524335946}{3681322727571460825} a^{19} + \frac{41413101700817216}{3681322727571460825} a^{18} - \frac{715831661930562}{147252909102858433} a^{17} - \frac{16839153557640313}{3681322727571460825} a^{16} + \frac{21512838491529629}{3681322727571460825} a^{15} - \frac{35809185018908003}{3681322727571460825} a^{14} - \frac{666859768224242}{10989023067377495} a^{13} - \frac{244570953406469744}{3681322727571460825} a^{12} + \frac{248914747283739543}{3681322727571460825} a^{11} + \frac{701818270132925823}{3681322727571460825} a^{10} - \frac{874157029916885644}{3681322727571460825} a^{9} + \frac{749621252316168962}{3681322727571460825} a^{8} + \frac{352205805577207763}{3681322727571460825} a^{7} + \frac{1262900245952788144}{3681322727571460825} a^{6} - \frac{708156518630675491}{3681322727571460825} a^{5} + \frac{542386315004721933}{3681322727571460825} a^{4} - \frac{30858635349612524}{99495208853282725} a^{3} + \frac{1329485317570882089}{3681322727571460825} a^{2} + \frac{208022044752082188}{736264545514292165} a - \frac{2411577070542585}{147252909102858433}$

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:  \( 193328971319.57028 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{13}\cdot 193328971319.57028 \cdot 1}{2\sqrt{1674969840842867253564530506487708596884007}}\approx 1.77664941527629$ (assuming GRH)

Galois group

$D_{26}$ (as 26T3):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 52
The 16 conjugacy class representatives for $D_{26}$
Character table for $D_{26}$

Intermediate fields

\(\Q(\sqrt{-7}) \), 13.1.489163986649360075249.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 26 sibling: Deg 26

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/LocalNumberField/2.13.0.1}{13} }^{2}$ $26$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{13}$ R ${\href{/LocalNumberField/11.13.0.1}{13} }^{2}$ $26$ $26$ $26$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.13.0.1}{13} }^{2}$ $26$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{13}$ ${\href{/LocalNumberField/43.13.0.1}{13} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{13}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ $26$

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$7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
401Data not computed

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.401.2t1.a.a$1$ $ 401 $ \(\Q(\sqrt{401}) \) $C_2$ (as 2T1) $1$ $1$
1.2807.2t1.a.a$1$ $ 7 \cdot 401 $ \(\Q(\sqrt{-2807}) \) $C_2$ (as 2T1) $1$ $-1$
* 1.7.2t1.a.a$1$ $ 7 $ \(\Q(\sqrt{-7}) \) $C_2$ (as 2T1) $1$ $-1$
* 2.2807.26t3.a.c$2$ $ 7 \cdot 401 $ 26.0.1674969840842867253564530506487708596884007.1 $D_{26}$ (as 26T3) $1$ $0$
* 2.2807.13t2.a.a$2$ $ 7 \cdot 401 $ 13.1.489163986649360075249.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.2807.13t2.a.e$2$ $ 7 \cdot 401 $ 13.1.489163986649360075249.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.2807.26t3.a.e$2$ $ 7 \cdot 401 $ 26.0.1674969840842867253564530506487708596884007.1 $D_{26}$ (as 26T3) $1$ $0$
* 2.2807.26t3.a.b$2$ $ 7 \cdot 401 $ 26.0.1674969840842867253564530506487708596884007.1 $D_{26}$ (as 26T3) $1$ $0$
* 2.2807.26t3.a.d$2$ $ 7 \cdot 401 $ 26.0.1674969840842867253564530506487708596884007.1 $D_{26}$ (as 26T3) $1$ $0$
* 2.2807.26t3.a.a$2$ $ 7 \cdot 401 $ 26.0.1674969840842867253564530506487708596884007.1 $D_{26}$ (as 26T3) $1$ $0$
* 2.2807.13t2.a.c$2$ $ 7 \cdot 401 $ 13.1.489163986649360075249.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.2807.13t2.a.f$2$ $ 7 \cdot 401 $ 13.1.489163986649360075249.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.2807.13t2.a.b$2$ $ 7 \cdot 401 $ 13.1.489163986649360075249.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.2807.26t3.a.f$2$ $ 7 \cdot 401 $ 26.0.1674969840842867253564530506487708596884007.1 $D_{26}$ (as 26T3) $1$ $0$
* 2.2807.13t2.a.d$2$ $ 7 \cdot 401 $ 13.1.489163986649360075249.1 $D_{13}$ (as 13T2) $1$ $0$

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 *.