Properties

Label 12.0.181612683140625.1
Degree $12$
Signature $[0, 6]$
Discriminant $1.816\times 10^{14}$
Root discriminant $15.43$
Ramified primes $3, 5, 11$
Class number $2$
Class group $[2]$
Galois group $C_6\times S_3$ (as 12T18)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 3*x^10 - 8*x^9 + 22*x^8 + 13*x^7 + 5*x^6 + 2*x^5 + 8*x^4 + 11*x^3 + 8*x^2 - x + 1)
 
gp: K = bnfinit(x^12 - 3*x^10 - 8*x^9 + 22*x^8 + 13*x^7 + 5*x^6 + 2*x^5 + 8*x^4 + 11*x^3 + 8*x^2 - x + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -1, 8, 11, 8, 2, 5, 13, 22, -8, -3, 0, 1]);
 

\(x^{12} - 3 x^{10} - 8 x^{9} + 22 x^{8} + 13 x^{7} + 5 x^{6} + 2 x^{5} + 8 x^{4} + 11 x^{3} + 8 x^{2} - x + 1\)  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:  \(181612683140625\)\(\medspace = 3^{8}\cdot 5^{6}\cdot 11^{6}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $15.43$
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:  $3, 5, 11$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$\card{ \Aut(K/\Q) }$:  $6$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2}$, $\frac{1}{10} a^{10} - \frac{1}{5} a^{9} - \frac{1}{10} a^{7} + \frac{2}{5} a^{6} - \frac{2}{5} a^{5} + \frac{2}{5} a^{4} - \frac{1}{5} a^{3} - \frac{1}{5} a^{2} - \frac{3}{10} a - \frac{2}{5}$, $\frac{1}{2345150} a^{11} - \frac{8581}{2345150} a^{10} + \frac{230363}{2345150} a^{9} - \frac{721521}{2345150} a^{8} + \frac{175723}{2345150} a^{7} + \frac{151189}{469030} a^{6} - \frac{54817}{234515} a^{5} - \frac{27549}{1172575} a^{4} + \frac{5347}{13175} a^{3} + \frac{23071}{469030} a^{2} - \frac{1146007}{2345150} a + \frac{437601}{2345150}$  Toggle raw display

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

Class group and class number

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

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:  \( -1 \) (order $2$)  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:  \( 86.95395829502839 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{6}\cdot 86.95395829502839 \cdot 2}{2\sqrt{181612683140625}}\approx 0.397004545657124$

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

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{-55}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{5}, \sqrt{-11})\), 6.0.107811.1

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: 18.6.977228812651986017578125.1, 18.0.1300691549639793389396484375.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.6.0.1}{6} }^{2}$ R R ${\href{/padicField/7.6.0.1}{6} }^{2}$ R ${\href{/padicField/13.6.0.1}{6} }^{2}$ ${\href{/padicField/17.2.0.1}{2} }^{6}$ ${\href{/padicField/19.2.0.1}{2} }^{6}$ ${\href{/padicField/23.6.0.1}{6} }^{2}$ ${\href{/padicField/29.6.0.1}{6} }^{2}$ ${\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{6}$ ${\href{/padicField/37.6.0.1}{6} }^{2}$ ${\href{/padicField/41.6.0.1}{6} }^{2}$ ${\href{/padicField/43.6.0.1}{6} }^{2}$ ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{3}$ ${\href{/padicField/53.6.0.1}{6} }^{2}$ ${\href{/padicField/59.3.0.1}{3} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{6}$

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
$3$3.6.8.4$x^{6} + 18 x^{2} + 63$$3$$2$$8$$C_6$$[2]^{2}$
3.6.0.1$x^{6} - x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
$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$$[\ ]_{2}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$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}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
* 1.55.2t1.a.a$1$ $ 5 \cdot 11 $ \(\Q(\sqrt{-55}) \) $C_2$ (as 2T1) $1$ $-1$
* 1.11.2t1.a.a$1$ $ 11 $ \(\Q(\sqrt{-11}) \) $C_2$ (as 2T1) $1$ $-1$
* 1.5.2t1.a.a$1$ $ 5 $ \(\Q(\sqrt{5}) \) $C_2$ (as 2T1) $1$ $1$
1.45.6t1.a.a$1$ $ 3^{2} \cdot 5 $ 6.6.820125.1 $C_6$ (as 6T1) $0$ $1$
1.495.6t1.b.a$1$ $ 3^{2} \cdot 5 \cdot 11 $ 6.0.1091586375.3 $C_6$ (as 6T1) $0$ $-1$
1.495.6t1.b.b$1$ $ 3^{2} \cdot 5 \cdot 11 $ 6.0.1091586375.3 $C_6$ (as 6T1) $0$ $-1$
1.99.6t1.a.a$1$ $ 3^{2} \cdot 11 $ 6.0.8732691.1 $C_6$ (as 6T1) $0$ $-1$
1.45.6t1.a.b$1$ $ 3^{2} \cdot 5 $ 6.6.820125.1 $C_6$ (as 6T1) $0$ $1$
1.9.3t1.a.a$1$ $ 3^{2}$ \(\Q(\zeta_{9})^+\) $C_3$ (as 3T1) $0$ $1$
1.9.3t1.a.b$1$ $ 3^{2}$ \(\Q(\zeta_{9})^+\) $C_3$ (as 3T1) $0$ $1$
1.99.6t1.a.b$1$ $ 3^{2} \cdot 11 $ 6.0.8732691.1 $C_6$ (as 6T1) $0$ $-1$
2.891.3t2.b.a$2$ $ 3^{4} \cdot 11 $ 3.1.891.1 $S_3$ (as 3T2) $1$ $0$
2.22275.6t3.c.a$2$ $ 3^{4} \cdot 5^{2} \cdot 11 $ 6.0.1091586375.2 $D_{6}$ (as 6T3) $1$ $0$
* 2.99.6t5.a.a$2$ $ 3^{2} \cdot 11 $ 6.0.107811.1 $S_3\times C_3$ (as 6T5) $0$ $0$
* 2.2475.12t18.b.a$2$ $ 3^{2} \cdot 5^{2} \cdot 11 $ 12.0.181612683140625.1 $C_6\times S_3$ (as 12T18) $0$ $0$
* 2.2475.12t18.b.b$2$ $ 3^{2} \cdot 5^{2} \cdot 11 $ 12.0.181612683140625.1 $C_6\times S_3$ (as 12T18) $0$ $0$
* 2.99.6t5.a.b$2$ $ 3^{2} \cdot 11 $ 6.0.107811.1 $S_3\times C_3$ (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 *.