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Properties

Label 12.0.57001208768...5625.1
Degree $12$
Signature $[0, 6]$
Discriminant $3^{6}\cdot 5^{6}\cdot 7^{10}\cdot 11^{6}$
Root discriminant $65.01$
Ramified primes $3, 5, 7, 11$
Class number $1296$ (GRH)
Class group $[3, 12, 36]$ (GRH)
Galois group $C_6\times C_2$ (as 12T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![435799, -513709, 143129, -119413, 71134, -1666, 12769, -680, 1184, -36, 50, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 + 50*x^10 - 36*x^9 + 1184*x^8 - 680*x^7 + 12769*x^6 - 1666*x^5 + 71134*x^4 - 119413*x^3 + 143129*x^2 - 513709*x + 435799)
 
gp: K = bnfinit(x^12 - x^11 + 50*x^10 - 36*x^9 + 1184*x^8 - 680*x^7 + 12769*x^6 - 1666*x^5 + 71134*x^4 - 119413*x^3 + 143129*x^2 - 513709*x + 435799, 1)
 

Normalized defining polynomial

\( x^{12} - x^{11} + 50 x^{10} - 36 x^{9} + 1184 x^{8} - 680 x^{7} + 12769 x^{6} - 1666 x^{5} + 71134 x^{4} - 119413 x^{3} + 143129 x^{2} - 513709 x + 435799 \)

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

Invariants

Degree:  $12$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(5700120876856238765625=3^{6}\cdot 5^{6}\cdot 7^{10}\cdot 11^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $65.01$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $3, 5, 7, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1155=3\cdot 5\cdot 7\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{1155}(1,·)$, $\chi_{1155}(131,·)$, $\chi_{1155}(584,·)$, $\chi_{1155}(331,·)$, $\chi_{1155}(461,·)$, $\chi_{1155}(109,·)$, $\chi_{1155}(274,·)$, $\chi_{1155}(419,·)$, $\chi_{1155}(626,·)$, $\chi_{1155}(89,·)$, $\chi_{1155}(604,·)$, $\chi_{1155}(991,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2}$, $\frac{1}{26} a^{8} - \frac{2}{13} a^{7} - \frac{1}{13} a^{5} + \frac{6}{13} a^{4} + \frac{2}{13} a^{3} - \frac{11}{26} a$, $\frac{1}{104} a^{9} - \frac{1}{52} a^{8} + \frac{9}{52} a^{7} + \frac{11}{104} a^{6} + \frac{47}{104} a^{5} - \frac{11}{104} a^{4} + \frac{21}{104} a^{3} + \frac{1}{52} a^{2} + \frac{43}{104} a + \frac{1}{8}$, $\frac{1}{4413032} a^{10} + \frac{1251}{551629} a^{9} + \frac{37693}{2206516} a^{8} + \frac{254055}{4413032} a^{7} - \frac{1041435}{4413032} a^{6} - \frac{26805}{4413032} a^{5} - \frac{1386865}{4413032} a^{4} - \frac{90843}{1103258} a^{3} - \frac{947449}{4413032} a^{2} + \frac{583207}{4413032} a + \frac{10053}{169732}$, $\frac{1}{8129875494976614184} a^{11} + \frac{48025097641}{625375038075124168} a^{10} - \frac{7137953208077543}{2032468873744153546} a^{9} + \frac{141028838812504137}{8129875494976614184} a^{8} - \frac{239439881827951704}{1016234436872076773} a^{7} + \frac{823960129561861421}{4064937747488307092} a^{6} + \frac{882952521869013333}{2032468873744153546} a^{5} - \frac{403936328149716727}{8129875494976614184} a^{4} - \frac{3396644450456506083}{8129875494976614184} a^{3} - \frac{784027779737925397}{4064937747488307092} a^{2} - \frac{3477567521119493073}{8129875494976614184} a + \frac{10502476858261317}{78171879759390521}$

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

Class group and class number

$C_{3}\times C_{12}\times C_{36}$, which has order $1296$ (assuming GRH)

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $5$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
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)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 1378.5816793295 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_6$ (as 12T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 12
The 12 conjugacy class representatives for $C_6\times C_2$
Character table for $C_6\times C_2$

Intermediate fields

\(\Q(\sqrt{-231}) \), \(\Q(\sqrt{105}) \), \(\Q(\sqrt{-55}) \), \(\Q(\zeta_{7})^+\), \(\Q(\sqrt{-55}, \sqrt{105})\), 6.0.603993159.1, 6.6.56723625.1, 6.0.399466375.2

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 ${\href{/LocalNumberField/2.3.0.1}{3} }^{4}$ R R R R ${\href{/LocalNumberField/13.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{4}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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];
 
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]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$3$3.12.6.2$x^{12} + 108 x^{6} - 243 x^{2} + 2916$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$5$5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$7$7.6.5.2$x^{6} - 7$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.2$x^{6} - 7$$6$$1$$5$$C_6$$[\ ]_{6}$
$11$11.12.6.1$x^{12} + 242 x^{8} + 21296 x^{6} + 14641 x^{4} + 1932612 x^{2} + 113379904$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$