Properties

Label 12.0.153664000000.1
Degree $12$
Signature $[0, 6]$
Discriminant $153664000000$
Root discriminant $8.55$
Ramified primes $2, 5, 7$
Class number $1$
Class group trivial
Galois group $C_6\times S_3$ (as 12T18)

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Show commands for: SageMath / Pari/GP / Magma

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^12 + x^10 - 4*x^9 + 4*x^8 - 6*x^7 + 12*x^6 - 12*x^5 + 12*x^4 - 16*x^3 + 14*x^2 - 6*x + 1)
 
gp: K = bnfinit(x^12 + x^10 - 4*x^9 + 4*x^8 - 6*x^7 + 12*x^6 - 12*x^5 + 12*x^4 - 16*x^3 + 14*x^2 - 6*x + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -6, 14, -16, 12, -12, 12, -6, 4, -4, 1, 0, 1]);
 

\(x^{12} + x^{10} - 4 x^{9} + 4 x^{8} - 6 x^{7} + 12 x^{6} - 12 x^{5} + 12 x^{4} - 16 x^{3} + 14 x^{2} - 6 x + 1\)  Toggle raw display

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

Invariants

Degree:  $12$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 6]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(153664000000\)\(\medspace = 2^{12}\cdot 5^{6}\cdot 7^{4}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $8.55$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $2, 5, 7$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $6$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a$  Toggle raw display

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

Class group and class number

Trivial group, which has order $1$

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

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $5$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( \frac{14}{3} a^{11} + 3 a^{10} + 7 a^{9} - \frac{41}{3} a^{8} + \frac{32}{3} a^{7} - \frac{67}{3} a^{6} + \frac{124}{3} a^{5} - \frac{95}{3} a^{4} + \frac{116}{3} a^{3} - 51 a^{2} + \frac{107}{3} a - 9 \) (order $4$)  Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 5.58654010372 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{6}\cdot 5.58654010372 \cdot 1}{4\sqrt{153664000000}}\approx 0.219217930008$

Galois group

$C_6\times S_3$ (as 12T18):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 36
The 18 conjugacy class representatives for $C_6\times S_3$
Character table for $C_6\times S_3$

Intermediate fields

\(\Q(\sqrt{-5}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-1}) \), \(\Q(i, \sqrt{5})\), 6.0.392000.1

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

Sibling fields

Galois closure: Deg 36
Degree 18 siblings: 18.6.110730297608000000000.1, 18.0.56693912375296000000.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.6.0.1}{6} }{,}\,{\href{/padicField/3.2.0.1}{2} }^{3}$ R R ${\href{/padicField/11.6.0.1}{6} }^{2}$ ${\href{/padicField/13.2.0.1}{2} }^{6}$ ${\href{/padicField/17.6.0.1}{6} }^{2}$ ${\href{/padicField/19.6.0.1}{6} }^{2}$ ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{3}$ ${\href{/padicField/29.3.0.1}{3} }^{4}$ ${\href{/padicField/31.6.0.1}{6} }^{2}$ ${\href{/padicField/37.6.0.1}{6} }^{2}$ ${\href{/padicField/41.3.0.1}{3} }^{4}$ ${\href{/padicField/43.6.0.1}{6} }^{2}$ ${\href{/padicField/47.6.0.1}{6} }^{2}$ ${\href{/padicField/53.6.0.1}{6} }^{2}$ ${\href{/padicField/59.6.0.1}{6} }^{2}$

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

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$2.12.12.26$x^{12} - 162 x^{10} + 26423 x^{8} + 125508 x^{6} - 64481 x^{4} - 122498 x^{2} - 86071$$2$$6$$12$$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$$[\ ]_{2}^{3}$
$7$7.6.0.1$x^{6} + 3 x^{2} - x + 5$$1$$6$$0$$C_6$$[\ ]^{6}$
7.6.4.1$x^{6} + 35 x^{3} + 441$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
* 1.4.2t1.a.a$1$ $ 2^{2}$ \(\Q(\sqrt{-1}) \) $C_2$ (as 2T1) $1$ $-1$
* 1.20.2t1.a.a$1$ $ 2^{2} \cdot 5 $ \(\Q(\sqrt{-5}) \) $C_2$ (as 2T1) $1$ $-1$
* 1.5.2t1.a.a$1$ $ 5 $ \(\Q(\sqrt{5}) \) $C_2$ (as 2T1) $1$ $1$
1.7.3t1.a.a$1$ $ 7 $ \(\Q(\zeta_{7})^+\) $C_3$ (as 3T1) $0$ $1$
1.7.3t1.a.b$1$ $ 7 $ \(\Q(\zeta_{7})^+\) $C_3$ (as 3T1) $0$ $1$
1.140.6t1.b.a$1$ $ 2^{2} \cdot 5 \cdot 7 $ 6.0.19208000.1 $C_6$ (as 6T1) $0$ $-1$
1.140.6t1.b.b$1$ $ 2^{2} \cdot 5 \cdot 7 $ 6.0.19208000.1 $C_6$ (as 6T1) $0$ $-1$
1.35.6t1.b.a$1$ $ 5 \cdot 7 $ 6.6.300125.1 $C_6$ (as 6T1) $0$ $1$
1.28.6t1.a.a$1$ $ 2^{2} \cdot 7 $ 6.0.153664.1 $C_6$ (as 6T1) $0$ $-1$
1.28.6t1.a.b$1$ $ 2^{2} \cdot 7 $ 6.0.153664.1 $C_6$ (as 6T1) $0$ $-1$
1.35.6t1.b.b$1$ $ 5 \cdot 7 $ 6.6.300125.1 $C_6$ (as 6T1) $0$ $1$
2.980.3t2.a.a$2$ $ 2^{2} \cdot 5 \cdot 7^{2}$ 3.1.980.1 $S_3$ (as 3T2) $1$ $0$
2.980.6t3.e.a$2$ $ 2^{2} \cdot 5 \cdot 7^{2}$ 6.0.3841600.1 $D_{6}$ (as 6T3) $1$ $0$
* 2.140.12t18.a.a$2$ $ 2^{2} \cdot 5 \cdot 7 $ 12.0.153664000000.1 $C_6\times S_3$ (as 12T18) $0$ $0$
* 2.140.6t5.a.a$2$ $ 2^{2} \cdot 5 \cdot 7 $ 6.0.392000.1 $S_3\times C_3$ (as 6T5) $0$ $0$
* 2.140.12t18.a.b$2$ $ 2^{2} \cdot 5 \cdot 7 $ 12.0.153664000000.1 $C_6\times S_3$ (as 12T18) $0$ $0$
* 2.140.6t5.a.b$2$ $ 2^{2} \cdot 5 \cdot 7 $ 6.0.392000.1 $S_3\times C_3$ (as 6T5) $0$ $0$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.