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Properties

Label 12.0.72847063604...8125.1
Degree $12$
Signature $[0, 6]$
Discriminant $3^{6}\cdot 5^{9}\cdot 7^{8}\cdot 31^{6}$
Root discriminant $118.00$
Ramified primes $3, 5, 7, 31$
Class number $100048$ (GRH)
Class group $[2, 2, 25012]$ (GRH)
Galois group $C_{12}$ (as 12T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![24768500401, 799850077, 2825933888, 62972433, 134261761, 1585644, 3370404, 9178, 46713, -106, 338, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 + 338*x^10 - 106*x^9 + 46713*x^8 + 9178*x^7 + 3370404*x^6 + 1585644*x^5 + 134261761*x^4 + 62972433*x^3 + 2825933888*x^2 + 799850077*x + 24768500401)
 
gp: K = bnfinit(x^12 - x^11 + 338*x^10 - 106*x^9 + 46713*x^8 + 9178*x^7 + 3370404*x^6 + 1585644*x^5 + 134261761*x^4 + 62972433*x^3 + 2825933888*x^2 + 799850077*x + 24768500401, 1)
 

Normalized defining polynomial

\( x^{12} - x^{11} + 338 x^{10} - 106 x^{9} + 46713 x^{8} + 9178 x^{7} + 3370404 x^{6} + 1585644 x^{5} + 134261761 x^{4} + 62972433 x^{3} + 2825933888 x^{2} + 799850077 x + 24768500401 \)

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:  \(7284706360423786423828125=3^{6}\cdot 5^{9}\cdot 7^{8}\cdot 31^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $118.00$
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:  $3, 5, 7, 31$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3255=3\cdot 5\cdot 7\cdot 31\)
Dirichlet character group:    $\lbrace$$\chi_{3255}(1024,·)$, $\chi_{3255}(1,·)$, $\chi_{3255}(1954,·)$, $\chi_{3255}(743,·)$, $\chi_{3255}(92,·)$, $\chi_{3255}(557,·)$, $\chi_{3255}(2417,·)$, $\chi_{3255}(466,·)$, $\chi_{3255}(2419,·)$, $\chi_{3255}(2326,·)$, $\chi_{3255}(1208,·)$, $\chi_{3255}(3068,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{599} a^{9} + \frac{69}{599} a^{8} - \frac{57}{599} a^{7} + \frac{240}{599} a^{6} + \frac{179}{599} a^{5} + \frac{54}{599} a^{4} - \frac{191}{599} a^{3} - \frac{171}{599} a^{2} - \frac{258}{599} a + \frac{246}{599}$, $\frac{1}{599} a^{10} - \frac{26}{599} a^{8} - \frac{20}{599} a^{7} - \frac{208}{599} a^{6} + \frac{282}{599} a^{5} + \frac{276}{599} a^{4} - \frac{170}{599} a^{3} + \frac{160}{599} a^{2} + \frac{78}{599} a - \frac{202}{599}$, $\frac{1}{204875788627772913104149899141820651409} a^{11} - \frac{127498142261895330772002444683301331}{204875788627772913104149899141820651409} a^{10} + \frac{123419684022748107393574831723220066}{204875788627772913104149899141820651409} a^{9} - \frac{7405227090868959137452964141884952092}{204875788627772913104149899141820651409} a^{8} - \frac{2293261308633544703057773819991687092}{204875788627772913104149899141820651409} a^{7} - \frac{25717183859670915346712854390591946380}{204875788627772913104149899141820651409} a^{6} - \frac{19508364428734195625864061804005605866}{204875788627772913104149899141820651409} a^{5} + \frac{26030420619256374031010108838808940364}{204875788627772913104149899141820651409} a^{4} - \frac{70860648800182324409938929721338424346}{204875788627772913104149899141820651409} a^{3} - \frac{31500972895022768275194952876766821788}{204875788627772913104149899141820651409} a^{2} - \frac{20671570181600905628648843879125929683}{204875788627772913104149899141820651409} a - \frac{10241186799819252466529672706883023146}{204875788627772913104149899141820651409}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{25012}$, which has order $100048$ (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:  \( 104.882003477 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{12}$ (as 12T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 12
The 12 conjugacy class representatives for $C_{12}$
Character table for $C_{12}$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{7})^+\), 4.0.1081125.1, 6.6.300125.1

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 ${\href{/LocalNumberField/2.12.0.1}{12} }$ R R R ${\href{/LocalNumberField/11.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/17.12.0.1}{12} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/23.12.0.1}{12} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}$ R ${\href{/LocalNumberField/37.12.0.1}{12} }$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/47.12.0.1}{12} }$ ${\href{/LocalNumberField/53.12.0.1}{12} }$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{4}$

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$3.12.6.1$x^{12} - 243 x^{2} + 1458$$2$$6$$6$$C_{12}$$[\ ]_{2}^{6}$
$5$5.12.9.1$x^{12} - 10 x^{8} - 375 x^{4} - 2000$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
$7$7.12.8.1$x^{12} - 63 x^{9} + 637 x^{6} + 6174 x^{3} + 300125$$3$$4$$8$$C_{12}$$[\ ]_{3}^{4}$
$31$31.6.3.2$x^{6} - 961 x^{2} + 268119$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
31.6.3.2$x^{6} - 961 x^{2} + 268119$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$