Properties

Label 12.0.61132828589969773.1
Degree $12$
Signature $[0, 6]$
Discriminant $6.113\times 10^{16}$
Root discriminant $25.05$
Ramified primes $7, 13$
Class number $4$
Class group $[2, 2]$
Galois group $C_{12}$ (as 12T1)

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Show commands for: SageMath / Pari/GP / Magma

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 - 4*x^10 - 10*x^9 + 39*x^8 - 78*x^7 + 214*x^6 - 280*x^5 + 693*x^4 - 573*x^3 - 222*x^2 + 123*x + 601)
 
gp: K = bnfinit(x^12 - x^11 - 4*x^10 - 10*x^9 + 39*x^8 - 78*x^7 + 214*x^6 - 280*x^5 + 693*x^4 - 573*x^3 - 222*x^2 + 123*x + 601, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![601, 123, -222, -573, 693, -280, 214, -78, 39, -10, -4, -1, 1]);
 

\(x^{12} - x^{11} - 4 x^{10} - 10 x^{9} + 39 x^{8} - 78 x^{7} + 214 x^{6} - 280 x^{5} + 693 x^{4} - 573 x^{3} - 222 x^{2} + 123 x + 601\)  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:  \(61132828589969773\)\(\medspace = 7^{8}\cdot 13^{9}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $25.05$
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:  $7, 13$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Gal(K/\Q)|$:  $12$
This field is Galois and abelian over $\Q$.
Conductor:  \(91=7\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{91}(64,·)$, $\chi_{91}(1,·)$, $\chi_{91}(8,·)$, $\chi_{91}(44,·)$, $\chi_{91}(79,·)$, $\chi_{91}(18,·)$, $\chi_{91}(51,·)$, $\chi_{91}(53,·)$, $\chi_{91}(86,·)$, $\chi_{91}(25,·)$, $\chi_{91}(57,·)$, $\chi_{91}(60,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} + \frac{1}{3} a^{3} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{57088767249011397} a^{11} - \frac{1169367241308461}{19029589083003799} a^{10} - \frac{1610397135385953}{19029589083003799} a^{9} - \frac{7466144941468694}{19029589083003799} a^{8} + \frac{6376119561762270}{19029589083003799} a^{7} - \frac{8633952934697097}{19029589083003799} a^{6} + \frac{10512530373335212}{57088767249011397} a^{5} - \frac{3846971296255146}{19029589083003799} a^{4} - \frac{5120076750511115}{57088767249011397} a^{3} - \frac{15265976468250832}{57088767249011397} a^{2} + \frac{4682440286535021}{19029589083003799} a - \frac{5382215146921500}{19029589083003799}$  Toggle raw display

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

Class group and class number

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

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:  \( 562.775330001 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{6}\cdot 562.775330001 \cdot 4}{2\sqrt{61132828589969773}}\approx 0.280096067652$

Galois group

$C_{12}$ (as 12T1):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A cyclic group of order 12
The 12 conjugacy class representatives for $C_{12}$
Character table for $C_{12}$

Intermediate fields

\(\Q(\sqrt{13}) \), \(\Q(\zeta_{7})^+\), 4.0.2197.1, 6.6.5274997.1

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

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.12.0.1}{12} }$ ${\href{/padicField/3.3.0.1}{3} }^{4}$ ${\href{/padicField/5.12.0.1}{12} }$ R ${\href{/padicField/11.12.0.1}{12} }$ R ${\href{/padicField/17.6.0.1}{6} }^{2}$ ${\href{/padicField/19.12.0.1}{12} }$ ${\href{/padicField/23.6.0.1}{6} }^{2}$ ${\href{/padicField/29.1.0.1}{1} }^{12}$ ${\href{/padicField/31.12.0.1}{12} }$ ${\href{/padicField/37.12.0.1}{12} }$ ${\href{/padicField/41.4.0.1}{4} }^{3}$ ${\href{/padicField/43.2.0.1}{2} }^{6}$ ${\href{/padicField/47.12.0.1}{12} }$ ${\href{/padicField/53.3.0.1}{3} }^{4}$ ${\href{/padicField/59.12.0.1}{12} }$

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
$7$7.12.8.1$x^{12} - 63 x^{9} + 637 x^{6} + 6174 x^{3} + 300125$$3$$4$$8$$C_{12}$$[\ ]_{3}^{4}$
$13$13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
* 1.13.4t1.a.a$1$ $ 13 $ 4.0.2197.1 $C_4$ (as 4T1) $0$ $-1$
* 1.13.2t1.a.a$1$ $ 13 $ \(\Q(\sqrt{13}) \) $C_2$ (as 2T1) $1$ $1$
* 1.13.4t1.a.b$1$ $ 13 $ 4.0.2197.1 $C_4$ (as 4T1) $0$ $-1$
* 1.7.3t1.a.a$1$ $ 7 $ \(\Q(\zeta_{7})^+\) $C_3$ (as 3T1) $0$ $1$
* 1.91.12t1.a.a$1$ $ 7 \cdot 13 $ 12.0.61132828589969773.1 $C_{12}$ (as 12T1) $0$ $-1$
* 1.91.6t1.j.a$1$ $ 7 \cdot 13 $ 6.6.5274997.1 $C_6$ (as 6T1) $0$ $1$
* 1.91.12t1.a.b$1$ $ 7 \cdot 13 $ 12.0.61132828589969773.1 $C_{12}$ (as 12T1) $0$ $-1$
* 1.7.3t1.a.b$1$ $ 7 $ \(\Q(\zeta_{7})^+\) $C_3$ (as 3T1) $0$ $1$
* 1.91.12t1.a.c$1$ $ 7 \cdot 13 $ 12.0.61132828589969773.1 $C_{12}$ (as 12T1) $0$ $-1$
* 1.91.6t1.j.b$1$ $ 7 \cdot 13 $ 6.6.5274997.1 $C_6$ (as 6T1) $0$ $1$
* 1.91.12t1.a.d$1$ $ 7 \cdot 13 $ 12.0.61132828589969773.1 $C_{12}$ (as 12T1) $0$ $-1$

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 *.