Properties

Label 26.2.129...328.1
Degree $26$
Signature $[2, 12]$
Discriminant $1.296\times 10^{39}$
Root discriminant $31.94$
Ramified primes $2, 191$
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^24 + 24*x^22 - 16*x^20 + 176*x^18 - 608*x^16 + 320*x^14 + 3712*x^12 + 1280*x^10 - 17408*x^8 + 22528*x^6 + 98304*x^4 + 65536*x^2 - 8192)
 
gp: K = bnfinit(x^26 - 8*x^24 + 24*x^22 - 16*x^20 + 176*x^18 - 608*x^16 + 320*x^14 + 3712*x^12 + 1280*x^10 - 17408*x^8 + 22528*x^6 + 98304*x^4 + 65536*x^2 - 8192, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-8192, 0, 65536, 0, 98304, 0, 22528, 0, -17408, 0, 1280, 0, 3712, 0, 320, 0, -608, 0, 176, 0, -16, 0, 24, 0, -8, 0, 1]);
 

\(x^{26} - 8 x^{24} + 24 x^{22} - 16 x^{20} + 176 x^{18} - 608 x^{16} + 320 x^{14} + 3712 x^{12} + 1280 x^{10} - 17408 x^{8} + 22528 x^{6} + 98304 x^{4} + 65536 x^{2} - 8192\)  Toggle raw display

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:  \(1295896264146521980742254147120375267328\)\(\medspace = 2^{39}\cdot 191^{12}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $31.94$
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:  $2, 191$
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 \(a\))

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{4} a^{4}$, $\frac{1}{4} a^{5}$, $\frac{1}{8} a^{6}$, $\frac{1}{8} a^{7}$, $\frac{1}{16} a^{8}$, $\frac{1}{16} a^{9}$, $\frac{1}{32} a^{10}$, $\frac{1}{32} a^{11}$, $\frac{1}{64} a^{12}$, $\frac{1}{64} a^{13}$, $\frac{1}{128} a^{14}$, $\frac{1}{128} a^{15}$, $\frac{1}{256} a^{16}$, $\frac{1}{256} a^{17}$, $\frac{1}{512} a^{18}$, $\frac{1}{512} a^{19}$, $\frac{1}{7168} a^{20} + \frac{3}{3584} a^{18} + \frac{3}{1792} a^{16} - \frac{1}{896} a^{14} - \frac{3}{448} a^{12} - \frac{3}{112} a^{8} - \frac{1}{56} a^{6} + \frac{1}{7} a^{2} - \frac{2}{7}$, $\frac{1}{7168} a^{21} + \frac{3}{3584} a^{19} + \frac{3}{1792} a^{17} - \frac{1}{896} a^{15} - \frac{3}{448} a^{13} - \frac{3}{112} a^{9} - \frac{1}{56} a^{7} + \frac{1}{7} a^{3} - \frac{2}{7} a$, $\frac{1}{14336} a^{22} + \frac{1}{3584} a^{18} - \frac{3}{1792} a^{16} + \frac{1}{224} a^{12} - \frac{3}{224} a^{10} + \frac{1}{112} a^{8} + \frac{3}{56} a^{6} + \frac{1}{14} a^{4} - \frac{1}{14} a^{2} - \frac{1}{7}$, $\frac{1}{14336} a^{23} + \frac{1}{3584} a^{19} - \frac{3}{1792} a^{17} + \frac{1}{224} a^{13} - \frac{3}{224} a^{11} + \frac{1}{112} a^{9} + \frac{3}{56} a^{7} + \frac{1}{14} a^{5} - \frac{1}{14} a^{3} - \frac{1}{7} a$, $\frac{1}{8845098938368} a^{24} + \frac{144081863}{4422549469184} a^{22} - \frac{3042471}{138204670912} a^{20} - \frac{433108185}{1105637367296} a^{18} - \frac{43879651}{552818683648} a^{16} + \frac{44456947}{17275583864} a^{14} - \frac{8238465}{3735261376} a^{12} - \frac{813048479}{69102335456} a^{10} + \frac{1026038639}{34551167728} a^{8} - \frac{338755635}{8637791932} a^{6} - \frac{449180335}{4318895966} a^{4} + \frac{524958761}{2159447983} a^{2} + \frac{940912370}{2159447983}$, $\frac{1}{8845098938368} a^{25} + \frac{144081863}{4422549469184} a^{23} - \frac{3042471}{138204670912} a^{21} - \frac{433108185}{1105637367296} a^{19} - \frac{43879651}{552818683648} a^{17} + \frac{44456947}{17275583864} a^{15} - \frac{8238465}{3735261376} a^{13} - \frac{813048479}{69102335456} a^{11} + \frac{1026038639}{34551167728} a^{9} - \frac{338755635}{8637791932} a^{7} - \frac{449180335}{4318895966} a^{5} + \frac{524958761}{2159447983} a^{3} + \frac{940912370}{2159447983} a$  Toggle raw display

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:  $13$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)  Toggle raw display
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:  \( 1293801547.1375248 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{2}\cdot(2\pi)^{12}\cdot 1293801547.1375248 \cdot 1}{2\sqrt{1295896264146521980742254147120375267328}}\approx 0.272126579712830$ (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

\(\Q(\sqrt{2}) \), 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 $26$ $26$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{13}$ $26$ ${\href{/LocalNumberField/17.13.0.1}{13} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{13}$ ${\href{/LocalNumberField/23.13.0.1}{13} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{13}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{13}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ $26$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{13}$ $26$

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
2Data 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$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.1528.2t1.b.a$1$ $ 2^{3} \cdot 191 $ \(\Q(\sqrt{-382}) \) $C_2$ (as 2T1) $1$ $-1$
1.191.2t1.a.a$1$ $ 191 $ \(\Q(\sqrt{-191}) \) $C_2$ (as 2T1) $1$ $-1$
* 1.8.2t1.a.a$1$ $ 2^{3}$ \(\Q(\sqrt{2}) \) $C_2$ (as 2T1) $1$ $1$
* 2.12224.26t3.a.e$2$ $ 2^{6} \cdot 191 $ 26.2.1295896264146521980742254147120375267328.1 $D_{26}$ (as 26T3) $1$ $0$
* 2.191.13t2.a.f$2$ $ 191 $ 13.1.48551226272641.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.191.13t2.a.e$2$ $ 191 $ 13.1.48551226272641.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.12224.26t3.a.d$2$ $ 2^{6} \cdot 191 $ 26.2.1295896264146521980742254147120375267328.1 $D_{26}$ (as 26T3) $1$ $0$
* 2.191.13t2.a.b$2$ $ 191 $ 13.1.48551226272641.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.12224.26t3.a.b$2$ $ 2^{6} \cdot 191 $ 26.2.1295896264146521980742254147120375267328.1 $D_{26}$ (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.c$2$ $ 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) $1$ $0$
* 2.12224.26t3.a.c$2$ $ 2^{6} \cdot 191 $ 26.2.1295896264146521980742254147120375267328.1 $D_{26}$ (as 26T3) $1$ $0$
* 2.12224.26t3.a.f$2$ $ 2^{6} \cdot 191 $ 26.2.1295896264146521980742254147120375267328.1 $D_{26}$ (as 26T3) $1$ $0$
* 2.12224.26t3.a.a$2$ $ 2^{6} \cdot 191 $ 26.2.1295896264146521980742254147120375267328.1 $D_{26}$ (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 *.