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Properties

Label 26.0.308...216.1
Degree $26$
Signature $[0, 13]$
Discriminant $-3.082\times 10^{45}$
Root discriminant $56.18$
Ramified primes $2, 3, 263$
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 + 6*x^24 + 189*x^22 + 2700*x^20 + 22032*x^18 + 262683*x^16 + 3276126*x^14 + 20111652*x^12 + 46005732*x^10 - 38618046*x^8 - 195511239*x^6 - 5668704*x^4 + 416649744*x^2 + 419306949)
 
gp: K = bnfinit(x^26 + 6*x^24 + 189*x^22 + 2700*x^20 + 22032*x^18 + 262683*x^16 + 3276126*x^14 + 20111652*x^12 + 46005732*x^10 - 38618046*x^8 - 195511239*x^6 - 5668704*x^4 + 416649744*x^2 + 419306949, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![419306949, 0, 416649744, 0, -5668704, 0, -195511239, 0, -38618046, 0, 46005732, 0, 20111652, 0, 3276126, 0, 262683, 0, 22032, 0, 2700, 0, 189, 0, 6, 0, 1]);
 

\(x^{26} + 6 x^{24} + 189 x^{22} + 2700 x^{20} + 22032 x^{18} + 262683 x^{16} + 3276126 x^{14} + 20111652 x^{12} + 46005732 x^{10} - 38618046 x^{8} - 195511239 x^{6} - 5668704 x^{4} + 416649744 x^{2} + 419306949\)  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:  $[0, 13]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(-3081633091018542230201228818947762663997833216\)\(\medspace = -\,2^{26}\cdot 3^{13}\cdot 263^{13}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $56.18$
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:  $2, 3, 263$
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 \(a\))

$1$, $a$, $\frac{1}{3} a^{2}$, $\frac{1}{3} a^{3}$, $\frac{1}{9} a^{4}$, $\frac{1}{9} a^{5}$, $\frac{1}{27} a^{6}$, $\frac{1}{27} a^{7}$, $\frac{1}{81} a^{8}$, $\frac{1}{81} a^{9}$, $\frac{1}{243} a^{10}$, $\frac{1}{243} a^{11}$, $\frac{1}{729} a^{12}$, $\frac{1}{729} a^{13}$, $\frac{1}{10935} a^{14} - \frac{1}{3645} a^{12} + \frac{1}{1215} a^{10} + \frac{2}{405} a^{8} + \frac{2}{45} a^{4} - \frac{1}{5}$, $\frac{1}{10935} a^{15} - \frac{1}{3645} a^{13} + \frac{1}{1215} a^{11} + \frac{2}{405} a^{9} + \frac{2}{45} a^{5} - \frac{1}{5} a$, $\frac{1}{32805} a^{16} - \frac{2}{1215} a^{10} + \frac{2}{405} a^{8} + \frac{2}{135} a^{6} + \frac{2}{45} a^{4} - \frac{1}{15} a^{2} - \frac{1}{5}$, $\frac{1}{32805} a^{17} - \frac{2}{1215} a^{11} + \frac{2}{405} a^{9} + \frac{2}{135} a^{7} + \frac{2}{45} a^{5} - \frac{1}{15} a^{3} - \frac{1}{5} a$, $\frac{1}{98415} a^{18} - \frac{2}{3645} a^{12} + \frac{2}{1215} a^{10} + \frac{2}{405} a^{8} + \frac{2}{135} a^{6} - \frac{1}{45} a^{4} - \frac{1}{15} a^{2}$, $\frac{1}{98415} a^{19} - \frac{2}{3645} a^{13} + \frac{2}{1215} a^{11} + \frac{2}{405} a^{9} + \frac{2}{135} a^{7} - \frac{1}{45} a^{5} - \frac{1}{15} a^{3}$, $\frac{1}{295245} a^{20} - \frac{1}{1215} a^{10} + \frac{1}{405} a^{8} - \frac{1}{135} a^{6} - \frac{2}{45} a^{4} - \frac{2}{5}$, $\frac{1}{295245} a^{21} - \frac{1}{1215} a^{11} + \frac{1}{405} a^{9} - \frac{1}{135} a^{7} - \frac{2}{45} a^{5} - \frac{2}{5} a$, $\frac{1}{4428675} a^{22} + \frac{2}{54675} a^{14} + \frac{2}{18225} a^{12} - \frac{2}{6075} a^{10} - \frac{2}{2025} a^{8} - \frac{7}{675} a^{6} + \frac{4}{225} a^{4} - \frac{4}{25} a^{2} + \frac{3}{25}$, $\frac{1}{4428675} a^{23} + \frac{2}{54675} a^{15} + \frac{2}{18225} a^{13} - \frac{2}{6075} a^{11} - \frac{2}{2025} a^{9} - \frac{7}{675} a^{7} + \frac{4}{225} a^{5} - \frac{4}{25} a^{3} + \frac{3}{25} a$, $\frac{1}{9098119773341881190055102375} a^{24} - \frac{111517670754832745959}{3032706591113960396685034125} a^{22} - \frac{33576830419614384256}{202180439407597359779002275} a^{20} - \frac{110595241762464996469}{67393479802532453259667425} a^{18} + \frac{218371243523453242756}{16046066619650584109444625} a^{16} - \frac{1711053520226177381006}{37440822112518029588704125} a^{14} + \frac{3710581171935455428}{277339423055689108064475} a^{12} - \frac{2687234379156102847964}{4160091345835336620967125} a^{10} + \frac{5887616501728354162816}{1386697115278445540322375} a^{8} - \frac{2382698545381883999296}{154077457253160615591375} a^{6} + \frac{7426451603688192304507}{154077457253160615591375} a^{4} - \frac{4695904475112349667134}{51359152417720205197125} a^{2} - \frac{974655113857922722892}{17119717472573401732375}$, $\frac{1}{9098119773341881190055102375} a^{25} - \frac{111517670754832745959}{3032706591113960396685034125} a^{23} - \frac{33576830419614384256}{202180439407597359779002275} a^{21} - \frac{110595241762464996469}{67393479802532453259667425} a^{19} + \frac{218371243523453242756}{16046066619650584109444625} a^{17} - \frac{1711053520226177381006}{37440822112518029588704125} a^{15} + \frac{3710581171935455428}{277339423055689108064475} a^{13} - \frac{2687234379156102847964}{4160091345835336620967125} a^{11} + \frac{5887616501728354162816}{1386697115278445540322375} a^{9} - \frac{2382698545381883999296}{154077457253160615591375} a^{7} + \frac{7426451603688192304507}{154077457253160615591375} a^{5} - \frac{4695904475112349667134}{51359152417720205197125} a^{3} - \frac{974655113857922722892}{17119717472573401732375} a$  Toggle raw display

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:  $12$
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:  not computed  Toggle raw display
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

\(\Q(\sqrt{-789}) \), 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: 26.2.11717236087522974259320261669002899863109632.1

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 ${\href{/LocalNumberField/5.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/11.13.0.1}{13} }^{2}$ ${\href{/LocalNumberField/13.13.0.1}{13} }^{2}$ $26$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.13.0.1}{13} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ $26$ ${\href{/LocalNumberField/37.13.0.1}{13} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ $26$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{13}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{13}$

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
3Data not computed
263Data not computed

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
* 1.3156.2t1.a.a$1$ $ 2^{2} \cdot 3 \cdot 263 $ \(\Q(\sqrt{-789}) \) $C_2$ (as 2T1) $1$ $-1$
1.263.2t1.a.a$1$ $ 263 $ \(\Q(\sqrt{-263}) \) $C_2$ (as 2T1) $1$ $-1$
1.12.2t1.a.a$1$ $ 2^{2} \cdot 3 $ \(\Q(\sqrt{3}) \) $C_2$ (as 2T1) $1$ $1$
* 2.37872.26t3.a.f$2$ $ 2^{4} \cdot 3^{2} \cdot 263 $ 26.0.3081633091018542230201228818947762663997833216.1 $D_{26}$ (as 26T3) $1$ $0$
* 2.263.13t2.a.c$2$ $ 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) $1$ $0$
* 2.37872.26t3.a.e$2$ $ 2^{4} \cdot 3^{2} \cdot 263 $ 26.0.3081633091018542230201228818947762663997833216.1 $D_{26}$ (as 26T3) $1$ $0$
* 2.37872.26t3.a.b$2$ $ 2^{4} \cdot 3^{2} \cdot 263 $ 26.0.3081633091018542230201228818947762663997833216.1 $D_{26}$ (as 26T3) $1$ $0$
* 2.263.13t2.a.a$2$ $ 263 $ 13.1.330928743953809.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.263.13t2.a.d$2$ $ 263 $ 13.1.330928743953809.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.263.13t2.a.f$2$ $ 263 $ 13.1.330928743953809.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.37872.26t3.a.d$2$ $ 2^{4} \cdot 3^{2} \cdot 263 $ 26.0.3081633091018542230201228818947762663997833216.1 $D_{26}$ (as 26T3) $1$ $0$
* 2.37872.26t3.a.c$2$ $ 2^{4} \cdot 3^{2} \cdot 263 $ 26.0.3081633091018542230201228818947762663997833216.1 $D_{26}$ (as 26T3) $1$ $0$
* 2.263.13t2.a.e$2$ $ 263 $ 13.1.330928743953809.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.37872.26t3.a.a$2$ $ 2^{4} \cdot 3^{2} \cdot 263 $ 26.0.3081633091018542230201228818947762663997833216.1 $D_{26}$ (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 *.