Properties

Label 12.0.1200500000000.1
Degree $12$
Signature $[0, 6]$
Discriminant $1.200\times 10^{12}$
Root discriminant $10.15$
Ramified primes $2, 5, 7$
Class number $1$
Class group trivial
Galois group $S_3 \times C_4$ (as 12T11)

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Normalized defining polynomial

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

\(x^{12} - 2 x^{11} + 2 x^{10} - x^{9} + 5 x^{8} + 2 x^{7} - 3 x^{6} + 2 x^{5} + 5 x^{4} + 9 x^{3} + 7 x^{2} + 3 x + 1\)  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:  \(1200500000000\)\(\medspace = 2^{8}\cdot 5^{9}\cdot 7^{4}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $10.15$
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)));
 
$\card{ \Aut(K/\Q) }$:  $4$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{31} a^{11} + \frac{2}{31} a^{10} + \frac{10}{31} a^{9} + \frac{8}{31} a^{8} + \frac{6}{31} a^{7} - \frac{5}{31} a^{6} + \frac{8}{31} a^{5} + \frac{3}{31} a^{4} - \frac{14}{31} a^{3} + \frac{15}{31} a^{2} + \frac{5}{31} a - \frac{8}{31}$  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:  \( \frac{300}{31} a^{11} - \frac{826}{31} a^{10} + \frac{1233}{31} a^{9} - \frac{1258}{31} a^{8} + \frac{2482}{31} a^{7} - \frac{1283}{31} a^{6} + \frac{106}{31} a^{5} + \frac{528}{31} a^{4} + \frac{1039}{31} a^{3} + \frac{1989}{31} a^{2} + \frac{663}{31} a + \frac{452}{31} \) (order $10$)  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:  \( 44.4179208274 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{6}\cdot 44.4179208274 \cdot 1}{10\sqrt{1200500000000}}\approx 0.249434403396$

Galois group

$C_4\times S_3$ (as 12T11):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 24
The 12 conjugacy class representatives for $S_3 \times C_4$
Character table for $S_3 \times C_4$

Intermediate fields

\(\Q(\sqrt{5}) \), 3.1.140.1, \(\Q(\zeta_{5})\), 6.2.98000.1

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

Sibling fields

Galois closure: Deg 24
Degree 12 sibling: 12.4.58824500000000.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.12.0.1}{12} }$ R R ${\href{/padicField/11.3.0.1}{3} }^{4}$ ${\href{/padicField/13.12.0.1}{12} }$ ${\href{/padicField/17.12.0.1}{12} }$ ${\href{/padicField/19.2.0.1}{2} }^{6}$ ${\href{/padicField/23.4.0.1}{4} }^{3}$ ${\href{/padicField/29.6.0.1}{6} }^{2}$ ${\href{/padicField/31.2.0.1}{2} }^{4}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ ${\href{/padicField/37.4.0.1}{4} }^{3}$ ${\href{/padicField/41.2.0.1}{2} }^{4}{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ ${\href{/padicField/43.4.0.1}{4} }^{3}$ ${\href{/padicField/47.12.0.1}{12} }$ ${\href{/padicField/53.4.0.1}{4} }^{3}$ ${\href{/padicField/59.2.0.1}{2} }^{6}$

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
$2$2.12.8.1$x^{12} - 6 x^{9} + 12 x^{6} - 8 x^{3} + 16$$3$$4$$8$$C_3 : C_4$$[\ ]_{3}^{4}$
$5$5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$7$7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.35.2t1.a.a$1$ $ 5 \cdot 7 $ \(\Q(\sqrt{-35}) \) $C_2$ (as 2T1) $1$ $-1$
* 1.5.2t1.a.a$1$ $ 5 $ \(\Q(\sqrt{5}) \) $C_2$ (as 2T1) $1$ $1$
1.7.2t1.a.a$1$ $ 7 $ \(\Q(\sqrt{-7}) \) $C_2$ (as 2T1) $1$ $-1$
1.35.4t1.a.a$1$ $ 5 \cdot 7 $ 4.4.6125.1 $C_4$ (as 4T1) $0$ $1$
* 1.5.4t1.a.a$1$ $ 5 $ \(\Q(\zeta_{5})\) $C_4$ (as 4T1) $0$ $-1$
1.35.4t1.a.b$1$ $ 5 \cdot 7 $ 4.4.6125.1 $C_4$ (as 4T1) $0$ $1$
* 1.5.4t1.a.b$1$ $ 5 $ \(\Q(\zeta_{5})\) $C_4$ (as 4T1) $0$ $-1$
* 2.140.3t2.a.a$2$ $ 2^{2} \cdot 5 \cdot 7 $ 3.1.140.1 $S_3$ (as 3T2) $1$ $0$
* 2.140.6t3.b.a$2$ $ 2^{2} \cdot 5 \cdot 7 $ 6.2.98000.1 $D_{6}$ (as 6T3) $1$ $0$
* 2.700.12t11.a.a$2$ $ 2^{2} \cdot 5^{2} \cdot 7 $ 12.0.1200500000000.1 $S_3 \times C_4$ (as 12T11) $0$ $0$
* 2.700.12t11.a.b$2$ $ 2^{2} \cdot 5^{2} \cdot 7 $ 12.0.1200500000000.1 $S_3 \times C_4$ (as 12T11) $0$ $0$

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 *.