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Properties

Label 12.0.993958020055671489.4
Degree $12$
Signature $[0, 6]$
Discriminant $3^{6}\cdot 7^{10}\cdot 13^{6}$
Root discriminant $31.61$
Ramified primes $3, 7, 13$
Class number $16$
Class group $[2, 2, 4]$
Galois group $C_2^2 \times A_4$ (as 12T25)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1681, -2214, 6868, -2824, 3426, -1271, 1245, -376, 363, -74, 25, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 3*x^11 + 25*x^10 - 74*x^9 + 363*x^8 - 376*x^7 + 1245*x^6 - 1271*x^5 + 3426*x^4 - 2824*x^3 + 6868*x^2 - 2214*x + 1681)
 
gp: K = bnfinit(x^12 - 3*x^11 + 25*x^10 - 74*x^9 + 363*x^8 - 376*x^7 + 1245*x^6 - 1271*x^5 + 3426*x^4 - 2824*x^3 + 6868*x^2 - 2214*x + 1681, 1)
 

Normalized defining polynomial

\( x^{12} - 3 x^{11} + 25 x^{10} - 74 x^{9} + 363 x^{8} - 376 x^{7} + 1245 x^{6} - 1271 x^{5} + 3426 x^{4} - 2824 x^{3} + 6868 x^{2} - 2214 x + 1681 \)

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:  \(993958020055671489=3^{6}\cdot 7^{10}\cdot 13^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $31.61$
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:  $3, 7, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{58} a^{9} + \frac{23}{58} a^{7} - \frac{1}{58} a^{6} - \frac{5}{58} a^{5} + \frac{11}{29} a^{4} - \frac{17}{58} a^{3} - \frac{23}{58} a^{2} - \frac{1}{58} a - \frac{7}{58}$, $\frac{1}{580} a^{10} - \frac{1}{290} a^{9} - \frac{3}{290} a^{8} + \frac{49}{290} a^{7} + \frac{11}{116} a^{6} + \frac{177}{580} a^{5} + \frac{287}{580} a^{4} + \frac{243}{580} a^{3} - \frac{137}{290} a^{2} - \frac{1}{116} a - \frac{131}{580}$, $\frac{1}{573315397259736700} a^{11} - \frac{61417647553521}{114663079451947340} a^{10} - \frac{197947718071217}{28665769862986835} a^{9} - \frac{65070743203720147}{286657698629868350} a^{8} + \frac{270439464132868651}{573315397259736700} a^{7} - \frac{2224310843497708}{4942374114308075} a^{6} - \frac{73692765791344787}{286657698629868350} a^{5} - \frac{2288551353592089}{6991651186094350} a^{4} + \frac{279953141077310097}{573315397259736700} a^{3} + \frac{126583379986457707}{573315397259736700} a^{2} - \frac{35133120044639799}{143328849314934175} a - \frac{3846762582195417}{13983302372188700}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{4}$, which has order $16$

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:  \( 638.268269093 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times A_4$ (as 12T25):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 48
The 16 conjugacy class representatives for $C_2^2 \times A_4$
Character table for $C_2^2 \times A_4$

Intermediate fields

\(\Q(\sqrt{-91}) \), \(\Q(\zeta_{7})^+\), 6.4.842751.1, 6.0.36924979.1, 6.2.76690341.1

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

Sibling fields

Degree 12 siblings: 12.4.5881408402696281.1, 12.8.34801233152049.1, 12.4.5881408402696281.2, 12.0.993958020055671489.5
Degree 16 sibling: data not computed
Degree 24 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/5.3.0.1}{3} }^{4}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{4}$

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$$[\ ]_{2}^{6}$
$7$7.12.10.1$x^{12} - 70 x^{6} + 35721$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$
$13$13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$