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Properties

Label 12.0.93787457948...3125.1
Degree $12$
Signature $[0, 6]$
Discriminant $5^{9}\cdot 7^{8}\cdot 97^{6}$
Root discriminant $120.51$
Ramified primes $5, 7, 97$
Class number $176072$ (GRH)
Class group $[2, 2, 44018]$ (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![31603168001, 990002287, 3473412138, 74673288, 158746986, 1798584, 3828104, 9938, 50903, -111, 353, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 + 353*x^10 - 111*x^9 + 50903*x^8 + 9938*x^7 + 3828104*x^6 + 1798584*x^5 + 158746986*x^4 + 74673288*x^3 + 3473412138*x^2 + 990002287*x + 31603168001)
 
gp: K = bnfinit(x^12 - x^11 + 353*x^10 - 111*x^9 + 50903*x^8 + 9938*x^7 + 3828104*x^6 + 1798584*x^5 + 158746986*x^4 + 74673288*x^3 + 3473412138*x^2 + 990002287*x + 31603168001, 1)
 

Normalized defining polynomial

\( x^{12} - x^{11} + 353 x^{10} - 111 x^{9} + 50903 x^{8} + 9938 x^{7} + 3828104 x^{6} + 1798584 x^{5} + 158746986 x^{4} + 74673288 x^{3} + 3473412138 x^{2} + 990002287 x + 31603168001 \)

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:  \(9378745794895906501953125=5^{9}\cdot 7^{8}\cdot 97^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $120.51$
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:  $5, 7, 97$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3395=5\cdot 7\cdot 97\)
Dirichlet character group:    $\lbrace$$\chi_{3395}(1,·)$, $\chi_{3395}(387,·)$, $\chi_{3395}(389,·)$, $\chi_{3395}(193,·)$, $\chi_{3395}(872,·)$, $\chi_{3395}(1163,·)$, $\chi_{3395}(1359,·)$, $\chi_{3395}(1842,·)$, $\chi_{3395}(3299,·)$, $\chi_{3395}(1941,·)$, $\chi_{3395}(2426,·)$, $\chi_{3395}(3103,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{649} a^{9} + \frac{72}{649} a^{8} - \frac{1}{11} a^{7} + \frac{323}{649} a^{6} + \frac{8}{59} a^{5} - \frac{211}{649} a^{4} - \frac{311}{649} a^{3} - \frac{8}{649} a^{2} - \frac{1}{11} a + \frac{159}{649}$, $\frac{1}{649} a^{10} - \frac{51}{649} a^{8} + \frac{28}{649} a^{7} + \frac{196}{649} a^{6} - \frac{57}{649} a^{5} - \frac{46}{649} a^{4} + \frac{318}{649} a^{3} - \frac{12}{59} a^{2} - \frac{136}{649} a + \frac{234}{649}$, $\frac{1}{438267434827304989198319691755324038619} a^{11} - \frac{247020890495605353974061824833679754}{438267434827304989198319691755324038619} a^{10} - \frac{26209395321069793551938306993986583}{39842494075209544472574517432302185329} a^{9} + \frac{152139043971904386684316734004047379925}{438267434827304989198319691755324038619} a^{8} - \frac{90279167009369574864150825193138997121}{438267434827304989198319691755324038619} a^{7} + \frac{39320608685107470051338693506647156228}{438267434827304989198319691755324038619} a^{6} + \frac{82616636362791661253242159565729384116}{438267434827304989198319691755324038619} a^{5} - \frac{10467693392517706758226514231981218994}{39842494075209544472574517432302185329} a^{4} - \frac{10886078162464674697971516320600338620}{438267434827304989198319691755324038619} a^{3} + \frac{119843354873975789087578136828654999655}{438267434827304989198319691755324038619} a^{2} + \frac{129195463808982357713962399288310910518}{438267434827304989198319691755324038619} a - \frac{56398148751308110827929379888133413058}{438267434827304989198319691755324038619}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{44018}$, which has order $176072$ (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:  \( -1 \) (order $2$)
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:  \( 104.882003477 \) (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

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{7})^+\), 4.0.1176125.1, 6.6.300125.1

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 ${\href{/LocalNumberField/2.12.0.1}{12} }$ ${\href{/LocalNumberField/3.12.0.1}{12} }$ R R ${\href{/LocalNumberField/11.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/17.12.0.1}{12} }$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/23.12.0.1}{12} }$ ${\href{/LocalNumberField/29.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/37.12.0.1}{12} }$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/47.12.0.1}{12} }$ ${\href{/LocalNumberField/53.12.0.1}{12} }$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{4}$

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$5.12.9.1$x^{12} - 10 x^{8} - 375 x^{4} - 2000$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
$7$7.12.8.1$x^{12} - 63 x^{9} + 637 x^{6} + 6174 x^{3} + 300125$$3$$4$$8$$C_{12}$$[\ ]_{3}^{4}$
$97$97.4.2.2$x^{4} - 97 x^{2} + 47045$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
97.4.2.2$x^{4} - 97 x^{2} + 47045$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
97.4.2.2$x^{4} - 97 x^{2} + 47045$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$