↑

Properties

Label 12.4.12772361626...7797.4
Degree $12$
Signature $[4, 4]$
Discriminant $3^{16}\cdot 197^{5}$
Root discriminant $39.10$
Ramified primes $3, 197$
Class number $2$
Class group $[2]$
Galois group $(C_2^2.\SL(2,3)):C_2$ (as 12T104)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![17803, -6645, -17886, -1856, 2370, -663, -47, 240, -84, 9, 12, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 3*x^11 + 12*x^10 + 9*x^9 - 84*x^8 + 240*x^7 - 47*x^6 - 663*x^5 + 2370*x^4 - 1856*x^3 - 17886*x^2 - 6645*x + 17803)
 
gp: K = bnfinit(x^12 - 3*x^11 + 12*x^10 + 9*x^9 - 84*x^8 + 240*x^7 - 47*x^6 - 663*x^5 + 2370*x^4 - 1856*x^3 - 17886*x^2 - 6645*x + 17803, 1)
 

Normalized defining polynomial

\( x^{12} - 3 x^{11} + 12 x^{10} + 9 x^{9} - 84 x^{8} + 240 x^{7} - 47 x^{6} - 663 x^{5} + 2370 x^{4} - 1856 x^{3} - 17886 x^{2} - 6645 x + 17803 \)

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

Invariants

Degree:  $12$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(12772361626857247797=3^{16}\cdot 197^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $39.10$
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:  $3, 197$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{757} a^{10} - \frac{214}{757} a^{9} - \frac{320}{757} a^{8} - \frac{103}{757} a^{7} + \frac{377}{757} a^{6} + \frac{163}{757} a^{5} - \frac{276}{757} a^{4} - \frac{119}{757} a^{3} + \frac{275}{757} a^{2} + \frac{206}{757} a - \frac{17}{757}$, $\frac{1}{47616010425630341999} a^{11} + \frac{23732007285249672}{47616010425630341999} a^{10} - \frac{17120177974780012090}{47616010425630341999} a^{9} + \frac{20643641320978214950}{47616010425630341999} a^{8} - \frac{16588440323032829888}{47616010425630341999} a^{7} - \frac{6120990636415243132}{47616010425630341999} a^{6} + \frac{8346735935313034942}{47616010425630341999} a^{5} + \frac{47553724423488298}{47616010425630341999} a^{4} - \frac{6727180731907736455}{47616010425630341999} a^{3} - \frac{9627208617268197461}{47616010425630341999} a^{2} + \frac{9410122020855301353}{47616010425630341999} a + \frac{23414097309899623591}{47616010425630341999}$

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

Class group and class number

$C_{2}$, which has order $2$

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $7$
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:  \( 28091.1140467 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2^2.\SL(2,3)):C_2$ (as 12T104):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 192
The 18 conjugacy class representatives for $(C_2^2.\SL(2,3)):C_2$
Character table for $(C_2^2.\SL(2,3)):C_2$

Intermediate fields

\(\Q(\zeta_{9})^+\), 6.6.1292517.1

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

Sibling fields

Degree 12 siblings: 12.4.329108238471933.1, 12.4.329108238471933.2
Degree 24 siblings: data not computed
Arithmetically equvalently sibling: 12.4.12772361626857247797.3

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 ${\href{/LocalNumberField/5.12.0.1}{12} }$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/11.12.0.1}{12} }$ ${\href{/LocalNumberField/13.12.0.1}{12} }$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/31.12.0.1}{12} }$ ${\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{4}$

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$3.12.16.14$x^{12} + 72 x^{11} - 36 x^{10} + 108 x^{9} - 108 x^{8} + 54 x^{7} + 72 x^{6} - 81 x^{5} - 81 x^{4} - 81 x^{3} + 81 x^{2} - 81$$3$$4$$16$$C_{12}$$[2]^{4}$
197Data not computed