↑

Properties

Label 12.0.79310093272...3125.1
Degree $12$
Signature $[0, 6]$
Discriminant $5^{9}\cdot 67^{8}$
Root discriminant $55.16$
Ramified primes $5, 67$
Class number $52$ (GRH)
Class group $[2, 26]$ (GRH)
Galois group $C_{12}$ (as 12T1)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![625, 2750, 11975, 52015, 225921, 15193, 11262, 1330, 561, -50, 23, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 + 23*x^10 - 50*x^9 + 561*x^8 + 1330*x^7 + 11262*x^6 + 15193*x^5 + 225921*x^4 + 52015*x^3 + 11975*x^2 + 2750*x + 625)
 
gp: K = bnfinit(x^12 - x^11 + 23*x^10 - 50*x^9 + 561*x^8 + 1330*x^7 + 11262*x^6 + 15193*x^5 + 225921*x^4 + 52015*x^3 + 11975*x^2 + 2750*x + 625, 1)
 

Normalized defining polynomial

\( x^{12} - x^{11} + 23 x^{10} - 50 x^{9} + 561 x^{8} + 1330 x^{7} + 11262 x^{6} + 15193 x^{5} + 225921 x^{4} + 52015 x^{3} + 11975 x^{2} + 2750 x + 625 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $12$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(793100932727814453125=5^{9}\cdot 67^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $55.16$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $5, 67$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(335=5\cdot 67\)
Dirichlet character group:    $\lbrace$$\chi_{335}(96,·)$, $\chi_{335}(1,·)$, $\chi_{335}(163,·)$, $\chi_{335}(68,·)$, $\chi_{335}(37,·)$, $\chi_{335}(104,·)$, $\chi_{335}(297,·)$, $\chi_{335}(202,·)$, $\chi_{335}(171,·)$, $\chi_{335}(269,·)$, $\chi_{335}(238,·)$, $\chi_{335}(29,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} + \frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{1940136465} a^{9} - \frac{59497522}{646712155} a^{8} + \frac{460924813}{1940136465} a^{7} - \frac{33832556}{129342431} a^{6} - \frac{100521734}{1940136465} a^{5} + \frac{50184562}{129342431} a^{4} + \frac{426828982}{1940136465} a^{3} + \frac{121327091}{646712155} a^{2} - \frac{557995214}{1940136465} a + \frac{40735261}{129342431}$, $\frac{1}{9700682325} a^{10} - \frac{1}{9700682325} a^{9} - \frac{69844909}{3233560775} a^{8} - \frac{158126818}{388027293} a^{7} - \frac{218864513}{3233560775} a^{6} - \frac{259791149}{1940136465} a^{5} - \frac{1026977396}{3233560775} a^{4} - \frac{426828982}{9700682325} a^{3} + \frac{913412357}{3233560775} a^{2} + \frac{654837253}{1940136465} a + \frac{5072741}{388027293}$, $\frac{1}{48503411625} a^{11} - \frac{1}{48503411625} a^{10} - \frac{2}{48503411625} a^{9} + \frac{288485359}{1940136465} a^{8} - \frac{16830968639}{48503411625} a^{7} + \frac{3389476991}{9700682325} a^{6} + \frac{13241339737}{48503411625} a^{5} - \frac{6291206807}{48503411625} a^{4} - \frac{12494476754}{48503411625} a^{3} - \frac{2577230212}{9700682325} a^{2} + \frac{19544141}{646712155} a + \frac{160226464}{388027293}$

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

Class group and class number

$C_{2}\times C_{26}$, which has order $52$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $5$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{12592}{3233560775} a^{11} - \frac{289616}{3233560775} a^{10} + \frac{25184}{129342431} a^{9} - \frac{7064112}{3233560775} a^{8} + \frac{22348667}{3233560775} a^{7} - \frac{141811104}{3233560775} a^{6} - \frac{191310256}{3233560775} a^{5} - \frac{2844797232}{3233560775} a^{4} - \frac{130994576}{646712155} a^{3} - \frac{61150955744}{3233560775} a^{2} - \frac{1385120}{129342431} a - \frac{314800}{129342431} \) (order $10$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 22006.3183057 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

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

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

Intermediate fields

\(\Q(\sqrt{5}) \), 3.3.4489.1, \(\Q(\zeta_{5})\), 6.6.2518890125.1

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 ${\href{/LocalNumberField/2.12.0.1}{12} }$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{3}$ R ${\href{/LocalNumberField/7.12.0.1}{12} }$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/13.12.0.1}{12} }$ ${\href{/LocalNumberField/17.12.0.1}{12} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/23.12.0.1}{12} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/37.12.0.1}{12} }$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/47.12.0.1}{12} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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];
 
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]])
 

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$$[\ ]_{4}$
$67$67.12.8.1$x^{12} - 201 x^{9} + 13467 x^{6} - 300763 x^{3} + 161208968$$3$$4$$8$$C_{12}$$[\ ]_{3}^{4}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.5.4t1.1c1$1$ $ 5 $ $x^{4} - x^{3} + x^{2} - x + 1$ $C_4$ (as 4T1) $0$ $-1$
* 1.5.2t1.1c1$1$ $ 5 $ $x^{2} - x - 1$ $C_2$ (as 2T1) $1$ $1$
* 1.5.4t1.1c2$1$ $ 5 $ $x^{4} - x^{3} + x^{2} - x + 1$ $C_4$ (as 4T1) $0$ $-1$
* 1.67.3t1.1c1$1$ $ 67 $ $x^{3} - x^{2} - 22 x - 5$ $C_3$ (as 3T1) $0$ $1$
* 1.5_67.12t1.1c1$1$ $ 5 \cdot 67 $ $x^{12} - x^{11} + 23 x^{10} - 50 x^{9} + 561 x^{8} + 1330 x^{7} + 11262 x^{6} + 15193 x^{5} + 225921 x^{4} + 52015 x^{3} + 11975 x^{2} + 2750 x + 625$ $C_{12}$ (as 12T1) $0$ $-1$
* 1.5_67.6t1.1c1$1$ $ 5 \cdot 67 $ $x^{6} - x^{5} - 48 x^{4} + 57 x^{3} + 483 x^{2} - 626 x - 311$ $C_6$ (as 6T1) $0$ $1$
* 1.5_67.12t1.1c2$1$ $ 5 \cdot 67 $ $x^{12} - x^{11} + 23 x^{10} - 50 x^{9} + 561 x^{8} + 1330 x^{7} + 11262 x^{6} + 15193 x^{5} + 225921 x^{4} + 52015 x^{3} + 11975 x^{2} + 2750 x + 625$ $C_{12}$ (as 12T1) $0$ $-1$
* 1.67.3t1.1c2$1$ $ 67 $ $x^{3} - x^{2} - 22 x - 5$ $C_3$ (as 3T1) $0$ $1$
* 1.5_67.12t1.1c3$1$ $ 5 \cdot 67 $ $x^{12} - x^{11} + 23 x^{10} - 50 x^{9} + 561 x^{8} + 1330 x^{7} + 11262 x^{6} + 15193 x^{5} + 225921 x^{4} + 52015 x^{3} + 11975 x^{2} + 2750 x + 625$ $C_{12}$ (as 12T1) $0$ $-1$
* 1.5_67.6t1.1c2$1$ $ 5 \cdot 67 $ $x^{6} - x^{5} - 48 x^{4} + 57 x^{3} + 483 x^{2} - 626 x - 311$ $C_6$ (as 6T1) $0$ $1$
* 1.5_67.12t1.1c4$1$ $ 5 \cdot 67 $ $x^{12} - x^{11} + 23 x^{10} - 50 x^{9} + 561 x^{8} + 1330 x^{7} + 11262 x^{6} + 15193 x^{5} + 225921 x^{4} + 52015 x^{3} + 11975 x^{2} + 2750 x + 625$ $C_{12}$ (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 *.