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Properties

Label 26.2.252...432.1
Degree $26$
Signature $[2, 12]$
Discriminant $2.522\times 10^{41}$
Root discriminant $39.12$
Ramified primes $2, 3, 191$
Class number not computed
Class group not computed
Galois group $D_{26}$ (as 26T3)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^26 - 12*x^24 + 54*x^22 - 54*x^20 + 891*x^18 - 4617*x^16 + 3645*x^14 + 63423*x^12 + 32805*x^10 - 669222*x^8 + 1299078*x^6 + 8503056*x^4 + 8503056*x^2 - 1594323)
 
gp: K = bnfinit(x^26 - 12*x^24 + 54*x^22 - 54*x^20 + 891*x^18 - 4617*x^16 + 3645*x^14 + 63423*x^12 + 32805*x^10 - 669222*x^8 + 1299078*x^6 + 8503056*x^4 + 8503056*x^2 - 1594323, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1594323, 0, 8503056, 0, 8503056, 0, 1299078, 0, -669222, 0, 32805, 0, 63423, 0, 3645, 0, -4617, 0, 891, 0, -54, 0, 54, 0, -12, 0, 1]);
 

\( x^{26} - 12 x^{24} + 54 x^{22} - 54 x^{20} + 891 x^{18} - 4617 x^{16} + 3645 x^{14} + 63423 x^{12} + 32805 x^{10} - 669222 x^{8} + 1299078 x^{6} + 8503056 x^{4} + 8503056 x^{2} - 1594323 \)

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

Invariants

Degree:  $26$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[2, 12]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(252206691838729902820182233715746833170432\)\(\medspace = 2^{26}\cdot 3^{13}\cdot 191^{12}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $39.12$
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:  $2, 3, 191$
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 \(a\))

$1$, $a$, $\frac{1}{3} a^{2}$, $\frac{1}{3} a^{3}$, $\frac{1}{9} a^{4}$, $\frac{1}{9} a^{5}$, $\frac{1}{27} a^{6}$, $\frac{1}{27} a^{7}$, $\frac{1}{81} a^{8}$, $\frac{1}{81} a^{9}$, $\frac{1}{243} a^{10}$, $\frac{1}{243} a^{11}$, $\frac{1}{729} a^{12}$, $\frac{1}{729} a^{13}$, $\frac{1}{2187} a^{14}$, $\frac{1}{2187} a^{15}$, $\frac{1}{6561} a^{16}$, $\frac{1}{6561} a^{17}$, $\frac{1}{19683} a^{18}$, $\frac{1}{19683} a^{19}$, $\frac{1}{413343} a^{20} + \frac{1}{45927} a^{18} + \frac{1}{15309} a^{16} - \frac{1}{15309} a^{14} - \frac{1}{1701} a^{12} - \frac{1}{189} a^{8} - \frac{1}{189} a^{6} + \frac{2}{21} a^{2} - \frac{2}{7}$, $\frac{1}{413343} a^{21} + \frac{1}{45927} a^{19} + \frac{1}{15309} a^{17} - \frac{1}{15309} a^{15} - \frac{1}{1701} a^{13} - \frac{1}{189} a^{9} - \frac{1}{189} a^{7} + \frac{2}{21} a^{3} - \frac{2}{7} a$, $\frac{1}{1240029} a^{22} + \frac{1}{137781} a^{18} - \frac{1}{15309} a^{16} + \frac{2}{5103} a^{12} - \frac{1}{567} a^{10} + \frac{1}{567} a^{8} + \frac{1}{63} a^{6} + \frac{2}{63} a^{4} - \frac{1}{21} a^{2} - \frac{1}{7}$, $\frac{1}{1240029} a^{23} + \frac{1}{137781} a^{19} - \frac{1}{15309} a^{17} + \frac{2}{5103} a^{13} - \frac{1}{567} a^{11} + \frac{1}{567} a^{9} + \frac{1}{63} a^{7} + \frac{2}{63} a^{5} - \frac{1}{21} a^{3} - \frac{1}{7} a$, $\frac{1}{1147619195533503} a^{24} + \frac{144081863}{382539731844501} a^{22} - \frac{16226512}{42504414649389} a^{20} - \frac{144369395}{14168138216463} a^{18} - \frac{43879651}{14168138216463} a^{16} + \frac{711311152}{4722712738821} a^{14} - \frac{915385}{4727440179} a^{12} - \frac{813048479}{524745859869} a^{10} + \frac{1026038639}{174915286623} a^{8} - \frac{25093010}{2159447983} a^{6} - \frac{898360670}{19435031847} a^{4} + \frac{1049917522}{6478343949} a^{2} + \frac{940912370}{2159447983}$, $\frac{1}{1147619195533503} a^{25} + \frac{144081863}{382539731844501} a^{23} - \frac{16226512}{42504414649389} a^{21} - \frac{144369395}{14168138216463} a^{19} - \frac{43879651}{14168138216463} a^{17} + \frac{711311152}{4722712738821} a^{15} - \frac{915385}{4727440179} a^{13} - \frac{813048479}{524745859869} a^{11} + \frac{1026038639}{174915286623} a^{9} - \frac{25093010}{2159447983} a^{7} - \frac{898360670}{19435031847} a^{5} + \frac{1049917522}{6478343949} a^{3} + \frac{940912370}{2159447983} a$

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

Class group and class number

not computed

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $13$
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:  not computed
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  not computed
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) $ not computed

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

\(\Q(\sqrt{3}) \), 13.1.48551226272641.1

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 R R $26$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{13}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.13.0.1}{13} }^{2}$ $26$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{13}$ ${\href{/LocalNumberField/23.13.0.1}{13} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{13}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{13}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{13}$ $26$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{13}$ ${\href{/LocalNumberField/59.13.0.1}{13} }^{2}$

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
2Data not computed
3Data not computed
$191$$\Q_{191}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{191}$$x + 2$$1$$1$$0$Trivial$[\ ]$
191.2.1.2$x^{2} + 382$$2$$1$$1$$C_2$$[\ ]_{2}$
191.2.1.2$x^{2} + 382$$2$$1$$1$$C_2$$[\ ]_{2}$
191.2.1.2$x^{2} + 382$$2$$1$$1$$C_2$$[\ ]_{2}$
191.2.1.2$x^{2} + 382$$2$$1$$1$$C_2$$[\ ]_{2}$
191.2.1.2$x^{2} + 382$$2$$1$$1$$C_2$$[\ ]_{2}$
191.2.1.2$x^{2} + 382$$2$$1$$1$$C_2$$[\ ]_{2}$
191.2.1.2$x^{2} + 382$$2$$1$$1$$C_2$$[\ ]_{2}$
191.2.1.2$x^{2} + 382$$2$$1$$1$$C_2$$[\ ]_{2}$
191.2.1.2$x^{2} + 382$$2$$1$$1$$C_2$$[\ ]_{2}$
191.2.1.2$x^{2} + 382$$2$$1$$1$$C_2$$[\ ]_{2}$
191.2.1.2$x^{2} + 382$$2$$1$$1$$C_2$$[\ ]_{2}$
191.2.1.2$x^{2} + 382$$2$$1$$1$$C_2$$[\ ]_{2}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.2292.2t1.a.a$1$ $ 2^{2} \cdot 3 \cdot 191 $ \(\Q(\sqrt{-573}) \) $C_2$ (as 2T1) $1$ $-1$
1.191.2t1.a.a$1$ $ 191 $ \(\Q(\sqrt{-191}) \) $C_2$ (as 2T1) $1$ $-1$
* 1.12.2t1.a.a$1$ $ 2^{2} \cdot 3 $ \(\Q(\sqrt{3}) \) $C_2$ (as 2T1) $1$ $1$
* 2.27504.26t3.a.d$2$ $ 2^{4} \cdot 3^{2} \cdot 191 $ 26.2.252206691838729902820182233715746833170432.1 $D_{26}$ (as 26T3) $1$ $0$
* 2.191.13t2.a.e$2$ $ 191 $ 13.1.48551226272641.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.191.13t2.a.c$2$ $ 191 $ 13.1.48551226272641.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.27504.26t3.a.e$2$ $ 2^{4} \cdot 3^{2} \cdot 191 $ 26.2.252206691838729902820182233715746833170432.1 $D_{26}$ (as 26T3) $1$ $0$
* 2.27504.26t3.a.f$2$ $ 2^{4} \cdot 3^{2} \cdot 191 $ 26.2.252206691838729902820182233715746833170432.1 $D_{26}$ (as 26T3) $1$ $0$
* 2.191.13t2.a.d$2$ $ 191 $ 13.1.48551226272641.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.191.13t2.a.b$2$ $ 191 $ 13.1.48551226272641.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.191.13t2.a.a$2$ $ 191 $ 13.1.48551226272641.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.27504.26t3.a.a$2$ $ 2^{4} \cdot 3^{2} \cdot 191 $ 26.2.252206691838729902820182233715746833170432.1 $D_{26}$ (as 26T3) $1$ $0$
* 2.27504.26t3.a.c$2$ $ 2^{4} \cdot 3^{2} \cdot 191 $ 26.2.252206691838729902820182233715746833170432.1 $D_{26}$ (as 26T3) $1$ $0$
* 2.191.13t2.a.f$2$ $ 191 $ 13.1.48551226272641.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.27504.26t3.a.b$2$ $ 2^{4} \cdot 3^{2} \cdot 191 $ 26.2.252206691838729902820182233715746833170432.1 $D_{26}$ (as 26T3) $1$ $0$

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 *.