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Properties

Label 12.0.22579948934...0625.1
Degree $12$
Signature $[0, 6]$
Discriminant $5^{6}\cdot 11^{6}\cdot 13^{8}$
Root discriminant $41.00$
Ramified primes $5, 11, 13$
Class number $168$
Class group $[2, 2, 42]$
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![35495, -7080, 17649, -2724, 4471, -250, 707, -172, 103, 10, -1, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^11 - x^10 + 10*x^9 + 103*x^8 - 172*x^7 + 707*x^6 - 250*x^5 + 4471*x^4 - 2724*x^3 + 17649*x^2 - 7080*x + 35495)
 
gp: K = bnfinit(x^12 - 4*x^11 - x^10 + 10*x^9 + 103*x^8 - 172*x^7 + 707*x^6 - 250*x^5 + 4471*x^4 - 2724*x^3 + 17649*x^2 - 7080*x + 35495, 1)
 

Normalized defining polynomial

\( x^{12} - 4 x^{11} - x^{10} + 10 x^{9} + 103 x^{8} - 172 x^{7} + 707 x^{6} - 250 x^{5} + 4471 x^{4} - 2724 x^{3} + 17649 x^{2} - 7080 x + 35495 \)

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:  \(22579948934773140625=5^{6}\cdot 11^{6}\cdot 13^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $41.00$
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:  $5, 11, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(715=5\cdot 11\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{715}(1,·)$, $\chi_{715}(386,·)$, $\chi_{715}(131,·)$, $\chi_{715}(516,·)$, $\chi_{715}(549,·)$, $\chi_{715}(144,·)$, $\chi_{715}(529,·)$, $\chi_{715}(274,·)$, $\chi_{715}(419,·)$, $\chi_{715}(659,·)$, $\chi_{715}(406,·)$, $\chi_{715}(276,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a$, $\frac{1}{8} a^{9} + \frac{1}{8} a^{8} - \frac{1}{8} a^{6} + \frac{3}{8} a^{5} - \frac{1}{8} a^{4} + \frac{1}{8} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a + \frac{1}{8}$, $\frac{1}{7858040} a^{10} + \frac{10627}{392902} a^{9} + \frac{2575}{1571608} a^{8} - \frac{331249}{1571608} a^{7} - \frac{200484}{982255} a^{6} + \frac{50098}{196451} a^{5} + \frac{257555}{785804} a^{4} + \frac{122825}{1571608} a^{3} + \frac{697133}{3929020} a^{2} + \frac{274137}{1571608} a - \frac{256589}{1571608}$, $\frac{1}{659324484987800} a^{11} + \frac{2037046}{82415560623475} a^{10} - \frac{3407625844647}{131864896997560} a^{9} - \frac{9226898662007}{131864896997560} a^{8} - \frac{19798265209249}{82415560623475} a^{7} - \frac{69697031217473}{329662242493900} a^{6} - \frac{12769031747411}{32966224249390} a^{5} + \frac{39923201829927}{131864896997560} a^{4} - \frac{32834639079998}{82415560623475} a^{3} + \frac{1807096607353}{659324484987800} a^{2} + \frac{13001699169627}{131864896997560} a - \frac{38719205041}{2126853177380}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{42}$, which has order $168$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 615.54450504 \)
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

\(\Q(\sqrt{-11}) \), \(\Q(\sqrt{-55}) \), \(\Q(\sqrt{5}) \), 3.3.169.1, \(\Q(\sqrt{5}, \sqrt{-11})\), 6.0.38014691.1, 6.0.4751836375.3, 6.6.3570125.1

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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}$ R R ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{4}$

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
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
$11$11.6.3.2$x^{6} - 121 x^{2} + 3993$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
11.6.3.2$x^{6} - 121 x^{2} + 3993$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$13$13.6.4.3$x^{6} + 65 x^{3} + 1352$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
13.6.4.3$x^{6} + 65 x^{3} + 1352$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$