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Properties

Label 26.2.602...128.1
Degree $26$
Signature $[2, 12]$
Discriminant $6.021\times 10^{40}$
Root discriminant $37.02$
Ramified primes $2, 263$
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 - 12*x^24 + 44*x^22 - 24*x^20 - 272*x^18 + 1312*x^16 - 1408*x^14 - 3328*x^12 + 30208*x^10 - 19456*x^8 + 46080*x^6 + 204800*x^4 + 131072*x^2 - 8192)
 
gp: K = bnfinit(x^26 - 12*x^24 + 44*x^22 - 24*x^20 - 272*x^18 + 1312*x^16 - 1408*x^14 - 3328*x^12 + 30208*x^10 - 19456*x^8 + 46080*x^6 + 204800*x^4 + 131072*x^2 - 8192, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-8192, 0, 131072, 0, 204800, 0, 46080, 0, -19456, 0, 30208, 0, -3328, 0, -1408, 0, 1312, 0, -272, 0, -24, 0, 44, 0, -12, 0, 1]);
 

\( x^{26} - 12 x^{24} + 44 x^{22} - 24 x^{20} - 272 x^{18} + 1312 x^{16} - 1408 x^{14} - 3328 x^{12} + 30208 x^{10} - 19456 x^{8} + 46080 x^{6} + 204800 x^{4} + 131072 x^{2} - 8192 \)

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:  \(60205866708934265598847651067237790384128\)\(\medspace = 2^{39}\cdot 263^{12}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $37.02$
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:  $2, 263$
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 \(a\))

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{4} a^{4}$, $\frac{1}{4} a^{5}$, $\frac{1}{8} a^{6}$, $\frac{1}{8} a^{7}$, $\frac{1}{16} a^{8}$, $\frac{1}{16} a^{9}$, $\frac{1}{32} a^{10}$, $\frac{1}{32} a^{11}$, $\frac{1}{64} a^{12}$, $\frac{1}{64} a^{13}$, $\frac{1}{640} a^{14} + \frac{1}{160} a^{12} + \frac{1}{80} a^{8} + \frac{1}{20} a^{6} - \frac{1}{5} a^{2} + \frac{1}{5}$, $\frac{1}{640} a^{15} + \frac{1}{160} a^{13} + \frac{1}{80} a^{9} + \frac{1}{20} a^{7} - \frac{1}{5} a^{3} + \frac{1}{5} a$, $\frac{1}{1280} a^{16} + \frac{1}{320} a^{12} + \frac{1}{160} a^{10} + \frac{1}{40} a^{6} - \frac{1}{10} a^{4} - \frac{2}{5}$, $\frac{1}{1280} a^{17} + \frac{1}{320} a^{13} + \frac{1}{160} a^{11} + \frac{1}{40} a^{7} - \frac{1}{10} a^{5} - \frac{2}{5} a$, $\frac{1}{2560} a^{18} - \frac{1}{320} a^{12} + \frac{1}{40} a^{6} - \frac{1}{5}$, $\frac{1}{2560} a^{19} - \frac{1}{320} a^{13} + \frac{1}{40} a^{7} - \frac{1}{5} a$, $\frac{1}{5120} a^{20} + \frac{1}{160} a^{12} + \frac{1}{40} a^{8} + \frac{1}{20} a^{6} + \frac{1}{5} a^{2} + \frac{1}{5}$, $\frac{1}{5120} a^{21} + \frac{1}{160} a^{13} + \frac{1}{40} a^{9} + \frac{1}{20} a^{7} + \frac{1}{5} a^{3} + \frac{1}{5} a$, $\frac{1}{51200} a^{22} - \frac{1}{12800} a^{20} - \frac{1}{12800} a^{18} + \frac{1}{6400} a^{16} + \frac{1}{3200} a^{14} - \frac{1}{400} a^{12} - \frac{7}{800} a^{10} + \frac{1}{200} a^{8} + \frac{9}{200} a^{6} - \frac{3}{25} a^{2} - \frac{9}{25}$, $\frac{1}{51200} a^{23} - \frac{1}{12800} a^{21} - \frac{1}{12800} a^{19} + \frac{1}{6400} a^{17} + \frac{1}{3200} a^{15} - \frac{1}{400} a^{13} - \frac{7}{800} a^{11} + \frac{1}{200} a^{9} + \frac{9}{200} a^{7} - \frac{3}{25} a^{3} - \frac{9}{25} a$, $\frac{1}{4123689472000} a^{24} - \frac{15094789}{2061844736000} a^{22} - \frac{4680527}{1030922368000} a^{20} - \frac{8406689}{64432648000} a^{18} - \frac{18518299}{64432648000} a^{16} - \frac{24724933}{64432648000} a^{14} - \frac{79215211}{16108162000} a^{12} - \frac{321924899}{32216324000} a^{10} - \frac{5408989}{460233200} a^{8} - \frac{5475909}{4027040500} a^{6} + \frac{134919957}{2013520250} a^{4} + \frac{67403769}{1006760125} a^{2} + \frac{214299153}{1006760125}$, $\frac{1}{4123689472000} a^{25} - \frac{15094789}{2061844736000} a^{23} - \frac{4680527}{1030922368000} a^{21} - \frac{8406689}{64432648000} a^{19} - \frac{18518299}{64432648000} a^{17} - \frac{24724933}{64432648000} a^{15} - \frac{79215211}{16108162000} a^{13} - \frac{321924899}{32216324000} a^{11} - \frac{5408989}{460233200} a^{9} - \frac{5475909}{4027040500} a^{7} + \frac{134919957}{2013520250} a^{5} + \frac{67403769}{1006760125} a^{3} + \frac{214299153}{1006760125} a$

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 $2$)
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:  \( 12785934187.300812 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{2}\cdot(2\pi)^{12}\cdot 12785934187.300812 \cdot 1}{2\sqrt{60205866708934265598847651067237790384128}}\approx 0.394549479009867$ (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

\(\Q(\sqrt{2}) \), 13.1.330928743953809.1

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 $26$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{13}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ $26$ $26$ ${\href{/LocalNumberField/17.13.0.1}{13} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{13}$ ${\href{/LocalNumberField/23.13.0.1}{13} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{13}$ ${\href{/LocalNumberField/31.13.0.1}{13} }^{2}$ $26$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ $26$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{13}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{13}$

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
263Data not computed

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.2104.2t1.b.a$1$ $ 2^{3} \cdot 263 $ \(\Q(\sqrt{-526}) \) $C_2$ (as 2T1) $1$ $-1$
1.263.2t1.a.a$1$ $ 263 $ \(\Q(\sqrt{-263}) \) $C_2$ (as 2T1) $1$ $-1$
* 1.8.2t1.a.a$1$ $ 2^{3}$ \(\Q(\sqrt{2}) \) $C_2$ (as 2T1) $1$ $1$
* 2.16832.26t3.a.b$2$ $ 2^{6} \cdot 263 $ 26.2.60205866708934265598847651067237790384128.1 $D_{26}$ (as 26T3) $1$ $0$
* 2.263.13t2.a.e$2$ $ 263 $ 13.1.330928743953809.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.263.13t2.a.b$2$ $ 263 $ 13.1.330928743953809.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.16832.26t3.a.a$2$ $ 2^{6} \cdot 263 $ 26.2.60205866708934265598847651067237790384128.1 $D_{26}$ (as 26T3) $1$ $0$
* 2.16832.26t3.a.d$2$ $ 2^{6} \cdot 263 $ 26.2.60205866708934265598847651067237790384128.1 $D_{26}$ (as 26T3) $1$ $0$
* 2.16832.26t3.a.e$2$ $ 2^{6} \cdot 263 $ 26.2.60205866708934265598847651067237790384128.1 $D_{26}$ (as 26T3) $1$ $0$
* 2.16832.26t3.a.c$2$ $ 2^{6} \cdot 263 $ 26.2.60205866708934265598847651067237790384128.1 $D_{26}$ (as 26T3) $1$ $0$
* 2.263.13t2.a.d$2$ $ 263 $ 13.1.330928743953809.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.263.13t2.a.c$2$ $ 263 $ 13.1.330928743953809.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.263.13t2.a.a$2$ $ 263 $ 13.1.330928743953809.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.16832.26t3.a.f$2$ $ 2^{6} \cdot 263 $ 26.2.60205866708934265598847651067237790384128.1 $D_{26}$ (as 26T3) $1$ $0$
* 2.263.13t2.a.f$2$ $ 263 $ 13.1.330928743953809.1 $D_{13}$ (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 *.