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Properties

Label 12.0.48467925833...3125.2
Degree $12$
Signature $[0, 6]$
Discriminant $3^{16}\cdot 5^{9}\cdot 7^{8}$
Root discriminant $52.94$
Ramified primes $3, 5, 7$
Class number $444$ (GRH)
Class group $[2, 222]$ (GRH)
Galois group $C_{12}$ (as 12T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1500625, 900375, 540225, 281260, 143031, 30870, 10486, 2205, 441, -35, 21, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 + 21*x^10 - 35*x^9 + 441*x^8 + 2205*x^7 + 10486*x^6 + 30870*x^5 + 143031*x^4 + 281260*x^3 + 540225*x^2 + 900375*x + 1500625)
 
gp: K = bnfinit(x^12 + 21*x^10 - 35*x^9 + 441*x^8 + 2205*x^7 + 10486*x^6 + 30870*x^5 + 143031*x^4 + 281260*x^3 + 540225*x^2 + 900375*x + 1500625, 1)
 

Normalized defining polynomial

\( x^{12} + 21 x^{10} - 35 x^{9} + 441 x^{8} + 2205 x^{7} + 10486 x^{6} + 30870 x^{5} + 143031 x^{4} + 281260 x^{3} + 540225 x^{2} + 900375 x + 1500625 \)

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:  \(484679258335001953125=3^{16}\cdot 5^{9}\cdot 7^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $52.94$
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:  $3, 5, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(315=3^{2}\cdot 5\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{315}(256,·)$, $\chi_{315}(1,·)$, $\chi_{315}(67,·)$, $\chi_{315}(4,·)$, $\chi_{315}(193,·)$, $\chi_{315}(64,·)$, $\chi_{315}(268,·)$, $\chi_{315}(142,·)$, $\chi_{315}(79,·)$, $\chi_{315}(16,·)$, $\chi_{315}(253,·)$, $\chi_{315}(127,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{7} a^{3}$, $\frac{1}{7} a^{4}$, $\frac{1}{7} a^{5}$, $\frac{1}{49} a^{6}$, $\frac{1}{49} a^{7}$, $\frac{1}{49} a^{8}$, $\frac{1}{52086265} a^{9} + \frac{4859}{1488179} a^{8} - \frac{2407}{7440895} a^{7} + \frac{10925}{1488179} a^{6} + \frac{23159}{1062985} a^{5} + \frac{2413}{212597} a^{4} - \frac{60906}{1062985} a^{3} - \frac{7230}{30371} a^{2} + \frac{60667}{151855} a + \frac{189}{30371}$, $\frac{1}{260431325} a^{10} - \frac{1}{175} a^{8} - \frac{1}{245} a^{7} + \frac{3}{1225} a^{6} + \frac{5304}{96635} a^{5} + \frac{9}{175} a^{4} + \frac{1}{35} a^{3} + \frac{2}{25} a^{2} - \frac{1}{5} a + \frac{10195}{30371}$, $\frac{1}{1302156625} a^{11} - \frac{4}{1302156625} a^{9} + \frac{284274}{37204475} a^{8} + \frac{1072613}{186022375} a^{7} - \frac{242712}{37204475} a^{6} - \frac{92636}{26574625} a^{5} - \frac{222249}{5314925} a^{4} + \frac{1610319}{26574625} a^{3} - \frac{305161}{759275} a^{2} - \frac{20164}{151855} a + \frac{13333}{30371}$

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

Class group and class number

$C_{2}\times C_{222}$, which has order $444$ (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:  \( -\frac{1926}{1302156625} a^{11} + \frac{642}{260431325} a^{10} - \frac{5778}{186022375} a^{9} + \frac{3627}{37204475} a^{8} - \frac{137388}{186022375} a^{7} - \frac{11556}{5314925} a^{6} - \frac{267714}{26574625} a^{5} - \frac{105288}{5314925} a^{4} - \frac{4087819}{26574625} a^{3} - \frac{1926}{30371} a^{2} - \frac{3210}{30371} a \) (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:  \( 8321.40669675 \) (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

\(\Q(\sqrt{5}) \), 3.3.3969.2, \(\Q(\zeta_{5})\), 6.6.1969120125.2

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 ${\href{/LocalNumberField/2.12.0.1}{12} }$ R R R ${\href{/LocalNumberField/11.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/13.12.0.1}{12} }$ ${\href{/LocalNumberField/17.12.0.1}{12} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/37.12.0.1}{12} }$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/43.12.0.1}{12} }$ ${\href{/LocalNumberField/47.12.0.1}{12} }$ ${\href{/LocalNumberField/53.12.0.1}{12} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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

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
$3$3.12.16.25$x^{12} + 93 x^{11} - 36 x^{10} + 357 x^{9} + 270 x^{8} + 324 x^{7} + 207 x^{6} - 216 x^{5} - 324 x^{4} - 54 x^{3} - 81 x^{2} - 324$$3$$4$$16$$C_{12}$$[2]^{4}$
$5$5.4.3.2$x^{4} - 20$$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}$
$7$7.12.8.2$x^{12} + 49 x^{6} - 1029 x^{3} + 12005$$3$$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) $0$ $-1$
* 1.3e2_7.3t1.1c1$1$ $ 3^{2} \cdot 7 $ $x^{3} - 21 x - 35$ $C_3$ (as 3T1) $0$ $1$
* 1.3e2_5_7.12t1.1c1$1$ $ 3^{2} \cdot 5 \cdot 7 $ $x^{12} + 21 x^{10} - 35 x^{9} + 441 x^{8} + 2205 x^{7} + 10486 x^{6} + 30870 x^{5} + 143031 x^{4} + 281260 x^{3} + 540225 x^{2} + 900375 x + 1500625$ $C_{12}$ (as 12T1) $0$ $-1$
* 1.3e2_5_7.6t1.4c1$1$ $ 3^{2} \cdot 5 \cdot 7 $ $x^{6} - 3 x^{5} - 42 x^{4} + 19 x^{3} + 483 x^{2} + 732 x + 251$ $C_6$ (as 6T1) $0$ $1$
* 1.3e2_5_7.12t1.1c2$1$ $ 3^{2} \cdot 5 \cdot 7 $ $x^{12} + 21 x^{10} - 35 x^{9} + 441 x^{8} + 2205 x^{7} + 10486 x^{6} + 30870 x^{5} + 143031 x^{4} + 281260 x^{3} + 540225 x^{2} + 900375 x + 1500625$ $C_{12}$ (as 12T1) $0$ $-1$
* 1.3e2_7.3t1.1c2$1$ $ 3^{2} \cdot 7 $ $x^{3} - 21 x - 35$ $C_3$ (as 3T1) $0$ $1$
* 1.3e2_5_7.12t1.1c3$1$ $ 3^{2} \cdot 5 \cdot 7 $ $x^{12} + 21 x^{10} - 35 x^{9} + 441 x^{8} + 2205 x^{7} + 10486 x^{6} + 30870 x^{5} + 143031 x^{4} + 281260 x^{3} + 540225 x^{2} + 900375 x + 1500625$ $C_{12}$ (as 12T1) $0$ $-1$
* 1.3e2_5_7.6t1.4c2$1$ $ 3^{2} \cdot 5 \cdot 7 $ $x^{6} - 3 x^{5} - 42 x^{4} + 19 x^{3} + 483 x^{2} + 732 x + 251$ $C_6$ (as 6T1) $0$ $1$
* 1.3e2_5_7.12t1.1c4$1$ $ 3^{2} \cdot 5 \cdot 7 $ $x^{12} + 21 x^{10} - 35 x^{9} + 441 x^{8} + 2205 x^{7} + 10486 x^{6} + 30870 x^{5} + 143031 x^{4} + 281260 x^{3} + 540225 x^{2} + 900375 x + 1500625$ $C_{12}$ (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 *.