Properties

Label 26.0.161...184.1
Degree $26$
Signature $[0, 13]$
Discriminant $-1.620\times 10^{49}$
Root discriminant $78.10$
Ramified primes $2, 53$
Class number $14209$ (GRH)
Class group $[14209]$ (GRH)
Galois group $C_{26}$ (as 26T1)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^26 + 49*x^24 + 994*x^22 + 10915*x^20 + 71324*x^18 + 287620*x^16 + 721007*x^14 + 1111118*x^12 + 1019820*x^10 + 521449*x^8 + 129173*x^6 + 10874*x^4 + 236*x^2 + 1)
 
gp: K = bnfinit(x^26 + 49*x^24 + 994*x^22 + 10915*x^20 + 71324*x^18 + 287620*x^16 + 721007*x^14 + 1111118*x^12 + 1019820*x^10 + 521449*x^8 + 129173*x^6 + 10874*x^4 + 236*x^2 + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 236, 0, 10874, 0, 129173, 0, 521449, 0, 1019820, 0, 1111118, 0, 721007, 0, 287620, 0, 71324, 0, 10915, 0, 994, 0, 49, 0, 1]);
 

\( x^{26} + 49 x^{24} + 994 x^{22} + 10915 x^{20} + 71324 x^{18} + 287620 x^{16} + 721007 x^{14} + 1111118 x^{12} + 1019820 x^{10} + 521449 x^{8} + 129173 x^{6} + 10874 x^{4} + 236 x^{2} + 1 \)

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:  \(-16195738565069746760238624235485184480969630941184\)\(\medspace = -\,2^{26}\cdot 53^{24}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $78.10$
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:  $2, 53$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Gal(K/\Q)|$:  $26$
This field is Galois and abelian over $\Q$.
Conductor:  \(212=2^{2}\cdot 53\)
Dirichlet character group:    $\lbrace$$\chi_{212}(1,·)$, $\chi_{212}(195,·)$, $\chi_{212}(69,·)$, $\chi_{212}(47,·)$, $\chi_{212}(203,·)$, $\chi_{212}(13,·)$, $\chi_{212}(77,·)$, $\chi_{212}(15,·)$, $\chi_{212}(81,·)$, $\chi_{212}(205,·)$, $\chi_{212}(89,·)$, $\chi_{212}(153,·)$, $\chi_{212}(201,·)$, $\chi_{212}(155,·)$, $\chi_{212}(95,·)$, $\chi_{212}(97,·)$, $\chi_{212}(99,·)$, $\chi_{212}(119,·)$, $\chi_{212}(169,·)$, $\chi_{212}(107,·)$, $\chi_{212}(175,·)$, $\chi_{212}(49,·)$, $\chi_{212}(183,·)$, $\chi_{212}(121,·)$, $\chi_{212}(187,·)$, $\chi_{212}(63,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{23} a^{16} - \frac{8}{23} a^{14} - \frac{7}{23} a^{10} - \frac{1}{23} a^{8} - \frac{4}{23} a^{6} + \frac{3}{23} a^{4} - \frac{8}{23} a^{2} - \frac{7}{23}$, $\frac{1}{23} a^{17} - \frac{8}{23} a^{15} - \frac{7}{23} a^{11} - \frac{1}{23} a^{9} - \frac{4}{23} a^{7} + \frac{3}{23} a^{5} - \frac{8}{23} a^{3} - \frac{7}{23} a$, $\frac{1}{23} a^{18} + \frac{5}{23} a^{14} - \frac{7}{23} a^{12} - \frac{11}{23} a^{10} + \frac{11}{23} a^{8} - \frac{6}{23} a^{6} - \frac{7}{23} a^{4} - \frac{2}{23} a^{2} - \frac{10}{23}$, $\frac{1}{23} a^{19} + \frac{5}{23} a^{15} - \frac{7}{23} a^{13} - \frac{11}{23} a^{11} + \frac{11}{23} a^{9} - \frac{6}{23} a^{7} - \frac{7}{23} a^{5} - \frac{2}{23} a^{3} - \frac{10}{23} a$, $\frac{1}{23} a^{20} + \frac{10}{23} a^{14} - \frac{11}{23} a^{12} - \frac{1}{23} a^{8} - \frac{10}{23} a^{6} + \frac{6}{23} a^{4} + \frac{7}{23} a^{2} - \frac{11}{23}$, $\frac{1}{23} a^{21} + \frac{10}{23} a^{15} - \frac{11}{23} a^{13} - \frac{1}{23} a^{9} - \frac{10}{23} a^{7} + \frac{6}{23} a^{5} + \frac{7}{23} a^{3} - \frac{11}{23} a$, $\frac{1}{4698049} a^{22} + \frac{34636}{4698049} a^{20} - \frac{91025}{4698049} a^{18} + \frac{26821}{4698049} a^{16} + \frac{379524}{4698049} a^{14} - \frac{255456}{4698049} a^{12} + \frac{2052906}{4698049} a^{10} + \frac{76757}{4698049} a^{8} + \frac{91766}{204263} a^{6} - \frac{1859271}{4698049} a^{4} + \frac{1415985}{4698049} a^{2} + \frac{2119684}{4698049}$, $\frac{1}{4698049} a^{23} + \frac{34636}{4698049} a^{21} - \frac{91025}{4698049} a^{19} + \frac{26821}{4698049} a^{17} + \frac{379524}{4698049} a^{15} - \frac{255456}{4698049} a^{13} + \frac{2052906}{4698049} a^{11} + \frac{76757}{4698049} a^{9} + \frac{91766}{204263} a^{7} - \frac{1859271}{4698049} a^{5} + \frac{1415985}{4698049} a^{3} + \frac{2119684}{4698049} a$, $\frac{1}{1070793422227} a^{24} + \frac{13278}{1070793422227} a^{22} - \frac{14186684266}{1070793422227} a^{20} - \frac{8086400107}{1070793422227} a^{18} - \frac{2421860596}{1070793422227} a^{16} + \frac{358322083704}{1070793422227} a^{14} - \frac{394474164910}{1070793422227} a^{12} - \frac{207849877869}{1070793422227} a^{10} - \frac{85443923132}{1070793422227} a^{8} + \frac{42726301558}{1070793422227} a^{6} + \frac{316600844179}{1070793422227} a^{4} + \frac{428682709255}{1070793422227} a^{2} - \frac{460404076292}{1070793422227}$, $\frac{1}{1070793422227} a^{25} + \frac{13278}{1070793422227} a^{23} - \frac{14186684266}{1070793422227} a^{21} - \frac{8086400107}{1070793422227} a^{19} - \frac{2421860596}{1070793422227} a^{17} + \frac{358322083704}{1070793422227} a^{15} - \frac{394474164910}{1070793422227} a^{13} - \frac{207849877869}{1070793422227} a^{11} - \frac{85443923132}{1070793422227} a^{9} + \frac{42726301558}{1070793422227} a^{7} + \frac{316600844179}{1070793422227} a^{5} + \frac{428682709255}{1070793422227} a^{3} - \frac{460404076292}{1070793422227} a$

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

Class group and class number

$C_{14209}$, which has order $14209$ (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:  \( -\frac{810188}{4698049} a^{25} - \frac{39731738}{4698049} a^{23} - \frac{806884047}{4698049} a^{21} - \frac{8873772064}{4698049} a^{19} - \frac{58106052589}{4698049} a^{17} - \frac{234977446694}{4698049} a^{15} - \frac{591220114790}{4698049} a^{13} - \frac{915134997367}{4698049} a^{11} - \frac{36668303943}{204263} a^{9} - \frac{431189338643}{4698049} a^{7} - \frac{104790832849}{4698049} a^{5} - \frac{7803798990}{4698049} a^{3} - \frac{4489009}{204263} a \) (order $4$)
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:  \( 5382739421.971964 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{13}\cdot 5382739421.971964 \cdot 14209}{4\sqrt{16195738565069746760238624235485184480969630941184}}\approx 0.113017263873654$ (assuming GRH)

Galois group

$C_{26}$ (as 26T1):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A cyclic group of order 26
The 26 conjugacy class representatives for $C_{26}$
Character table for $C_{26}$ is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), 13.13.491258904256726154641.1

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

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R $26$ ${\href{/LocalNumberField/5.13.0.1}{13} }^{2}$ $26$ $26$ ${\href{/LocalNumberField/13.13.0.1}{13} }^{2}$ ${\href{/LocalNumberField/17.13.0.1}{13} }^{2}$ $26$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{13}$ ${\href{/LocalNumberField/29.13.0.1}{13} }^{2}$ $26$ ${\href{/LocalNumberField/37.13.0.1}{13} }^{2}$ ${\href{/LocalNumberField/41.13.0.1}{13} }^{2}$ $26$ $26$ R $26$

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
$53$53.13.12.1$x^{13} - 53$$13$$1$$12$$C_{13}$$[\ ]_{13}$
53.13.12.1$x^{13} - 53$$13$$1$$12$$C_{13}$$[\ ]_{13}$