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)

Related objects

Show commands for: SageMath / Pari/GP / Magma

Normalizeddefining 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]);

$$x^{12} - x^{11} + 5 x^{10} - 10 x^{9} + 31 x^{8} + 50 x^{7} + 84 x^{6} + 85 x^{5} + 201 x^{4} + 55 x^{3} + 15 x^{2} + 4 x + 1$$

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 $$a$$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{6781} a^{9} + \frac{675}{6781} a^{8} + \frac{1298}{6781} a^{7} + \frac{1401}{6781} a^{6} + \frac{3116}{6781} a^{5} + \frac{658}{6781} a^{4} + \frac{3385}{6781} a^{3} - \frac{322}{6781} a^{2} - \frac{358}{6781} a + \frac{2466}{6781}$, $\frac{1}{6781} a^{10} - \frac{532}{6781} a^{5} - \frac{3205}{6781}$, $\frac{1}{6781} a^{11} - \frac{532}{6781} a^{6} - \frac{3205}{6781} a$

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 $10$) 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

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$$[\ ]_{4} 5.4.3.2x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4} 1313.12.8.1x^{12} - 39 x^{9} - 338 x^{6} + 10985 x^{3} + 228488$$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 *.