Properties

Label 12.0.1593224064453125.1
Degree $12$
Signature $[0, 6]$
Discriminant $1.593\times 10^{15}$
Root discriminant $18.49$
Ramified primes $5, 13$
Class number $4$
Class group $[2, 2]$
Galois group $C_{12}$ (as 12T1)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 + 5*x^10 - 10*x^9 + 31*x^8 + 50*x^7 + 84*x^6 + 85*x^5 + 201*x^4 + 55*x^3 + 15*x^2 + 4*x + 1)
 
gp: K = bnfinit(x^12 - x^11 + 5*x^10 - 10*x^9 + 31*x^8 + 50*x^7 + 84*x^6 + 85*x^5 + 201*x^4 + 55*x^3 + 15*x^2 + 4*x + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 4, 15, 55, 201, 85, 84, 50, 31, -10, 5, -1, 1]);
 

\(x^{12} - x^{11} + 5 x^{10} - 10 x^{9} + 31 x^{8} + 50 x^{7} + 84 x^{6} + 85 x^{5} + 201 x^{4} + 55 x^{3} + 15 x^{2} + 4 x + 1\)  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:  \(1593224064453125\)\(\medspace = 5^{9}\cdot 13^{8}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $18.49$
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:  $5, 13$
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:  \(65=5\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{65}(1,·)$, $\chi_{65}(3,·)$, $\chi_{65}(16,·)$, $\chi_{65}(9,·)$, $\chi_{65}(42,·)$, $\chi_{65}(14,·)$, $\chi_{65}(29,·)$, $\chi_{65}(48,·)$, $\chi_{65}(53,·)$, $\chi_{65}(22,·)$, $\chi_{65}(27,·)$, $\chi_{65}(61,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{6781} a^{9} + \frac{675}{6781} a^{8} + \frac{1298}{6781} a^{7} + \frac{1401}{6781} a^{6} + \frac{3116}{6781} a^{5} + \frac{658}{6781} a^{4} + \frac{3385}{6781} a^{3} - \frac{322}{6781} a^{2} - \frac{358}{6781} a + \frac{2466}{6781}$, $\frac{1}{6781} a^{10} - \frac{532}{6781} a^{5} - \frac{3205}{6781}$, $\frac{1}{6781} a^{11} - \frac{532}{6781} a^{6} - \frac{3205}{6781} a$  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:  \( \frac{134}{6781} a^{11} - \frac{670}{6781} a^{10} + \frac{1340}{6781} a^{9} - \frac{4154}{6781} a^{8} + \frac{10165}{6781} a^{7} - \frac{11256}{6781} a^{6} - \frac{11390}{6781} a^{5} - \frac{26934}{6781} a^{4} - \frac{7370}{6781} a^{3} - \frac{92430}{6781} a^{2} - \frac{536}{6781} a - \frac{134}{6781} \) (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:  \( 615.54450504 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{6}\cdot 615.54450504 \cdot 4}{10\sqrt{1593224064453125}}\approx 0.37954234055$

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

\(\Q(\sqrt{5}) \), 3.3.169.1, \(\Q(\zeta_{5})\), 6.6.3570125.1

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 ${\href{/padicField/2.12.0.1}{12} }$ ${\href{/padicField/3.12.0.1}{12} }$ R ${\href{/padicField/7.12.0.1}{12} }$ ${\href{/padicField/11.3.0.1}{3} }^{4}$ R ${\href{/padicField/17.12.0.1}{12} }$ ${\href{/padicField/19.6.0.1}{6} }^{2}$ ${\href{/padicField/23.12.0.1}{12} }$ ${\href{/padicField/29.6.0.1}{6} }^{2}$ ${\href{/padicField/31.1.0.1}{1} }^{12}$ ${\href{/padicField/37.12.0.1}{12} }$ ${\href{/padicField/41.3.0.1}{3} }^{4}$ ${\href{/padicField/43.12.0.1}{12} }$ ${\href{/padicField/47.4.0.1}{4} }^{3}$ ${\href{/padicField/53.4.0.1}{4} }^{3}$ ${\href{/padicField/59.6.0.1}{6} }^{2}$

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
$5$5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
$13$13.12.8.1$x^{12} - 39 x^{9} - 338 x^{6} + 10985 x^{3} + 228488$$3$$4$$8$$C_{12}$$[\ ]_{3}^{4}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
* 1.5.4t1.a.a$1$ $ 5 $ \(\Q(\zeta_{5})\) $C_4$ (as 4T1) $0$ $-1$
* 1.5.2t1.a.a$1$ $ 5 $ \(\Q(\sqrt{5}) \) $C_2$ (as 2T1) $1$ $1$
* 1.5.4t1.a.b$1$ $ 5 $ \(\Q(\zeta_{5})\) $C_4$ (as 4T1) $0$ $-1$
* 1.13.3t1.a.a$1$ $ 13 $ 3.3.169.1 $C_3$ (as 3T1) $0$ $1$
* 1.65.12t1.a.a$1$ $ 5 \cdot 13 $ 12.0.1593224064453125.1 $C_{12}$ (as 12T1) $0$ $-1$
* 1.65.6t1.b.a$1$ $ 5 \cdot 13 $ 6.6.3570125.1 $C_6$ (as 6T1) $0$ $1$
* 1.65.12t1.a.b$1$ $ 5 \cdot 13 $ 12.0.1593224064453125.1 $C_{12}$ (as 12T1) $0$ $-1$
* 1.13.3t1.a.b$1$ $ 13 $ 3.3.169.1 $C_3$ (as 3T1) $0$ $1$
* 1.65.12t1.a.c$1$ $ 5 \cdot 13 $ 12.0.1593224064453125.1 $C_{12}$ (as 12T1) $0$ $-1$
* 1.65.6t1.b.b$1$ $ 5 \cdot 13 $ 6.6.3570125.1 $C_6$ (as 6T1) $0$ $1$
* 1.65.12t1.a.d$1$ $ 5 \cdot 13 $ 12.0.1593224064453125.1 $C_{12}$ (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 *.