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Properties

Label 12.0.52834581635...3089.1
Degree $12$
Signature $[0, 6]$
Discriminant $3^{6}\cdot 7^{10}\cdot 37^{6}$
Root discriminant $53.32$
Ramified primes $3, 7, 37$
Class number $1008$ (GRH)
Class group $[6, 168]$ (GRH)
Galois group $C_6\times C_2$ (as 12T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1930411, 1079414, 822528, 187264, 377279, -5985, 55630, -1548, 3536, -71, 99, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 + 99*x^10 - 71*x^9 + 3536*x^8 - 1548*x^7 + 55630*x^6 - 5985*x^5 + 377279*x^4 + 187264*x^3 + 822528*x^2 + 1079414*x + 1930411)
 
gp: K = bnfinit(x^12 - x^11 + 99*x^10 - 71*x^9 + 3536*x^8 - 1548*x^7 + 55630*x^6 - 5985*x^5 + 377279*x^4 + 187264*x^3 + 822528*x^2 + 1079414*x + 1930411, 1)
 

Normalized defining polynomial

\( x^{12} - x^{11} + 99 x^{10} - 71 x^{9} + 3536 x^{8} - 1548 x^{7} + 55630 x^{6} - 5985 x^{5} + 377279 x^{4} + 187264 x^{3} + 822528 x^{2} + 1079414 x + 1930411 \)

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:  \(528345816354901963089=3^{6}\cdot 7^{10}\cdot 37^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $53.32$
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:  $3, 7, 37$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(777=3\cdot 7\cdot 37\)
Dirichlet character group:    $\lbrace$$\chi_{777}(1,·)$, $\chi_{777}(482,·)$, $\chi_{777}(667,·)$, $\chi_{777}(517,·)$, $\chi_{777}(38,·)$, $\chi_{777}(73,·)$, $\chi_{777}(554,·)$, $\chi_{777}(221,·)$, $\chi_{777}(593,·)$, $\chi_{777}(628,·)$, $\chi_{777}(443,·)$, $\chi_{777}(445,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{1261} a^{7} + \frac{63}{1261} a^{5} - \frac{127}{1261} a^{3} + \frac{59}{1261} a - \frac{2}{1261}$, $\frac{1}{1261} a^{8} + \frac{63}{1261} a^{6} - \frac{127}{1261} a^{4} + \frac{59}{1261} a^{2} - \frac{2}{1261} a$, $\frac{1}{12610} a^{9} + \frac{1}{12610} a^{7} - \frac{1}{10} a^{6} - \frac{4033}{12610} a^{5} - \frac{2}{5} a^{4} + \frac{5411}{12610} a^{3} + \frac{3781}{12610} a^{2} + \frac{693}{6305} a - \frac{3659}{12610}$, $\frac{1}{56096303770} a^{10} + \frac{188501}{56096303770} a^{9} - \frac{19026119}{56096303770} a^{8} + \frac{2197498}{5609630377} a^{7} - \frac{12692391087}{28048151885} a^{6} - \frac{6988670367}{56096303770} a^{5} - \frac{18807119433}{56096303770} a^{4} - \frac{10711650384}{28048151885} a^{3} - \frac{11337542533}{56096303770} a^{2} - \frac{9111670093}{56096303770} a - \frac{12701148999}{56096303770}$, $\frac{1}{56096303770} a^{11} + \frac{607212}{28048151885} a^{9} - \frac{6706371}{56096303770} a^{8} - \frac{647902}{5609630377} a^{7} - \frac{23961404267}{56096303770} a^{6} + \frac{10898995646}{28048151885} a^{5} + \frac{12354884019}{56096303770} a^{4} + \frac{8075994379}{56096303770} a^{3} - \frac{7939357183}{28048151885} a^{2} + \frac{3565216714}{28048151885} a + \frac{26419332813}{56096303770}$

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

Class group and class number

$C_{6}\times C_{168}$, which has order $1008$ (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:  \( 140.7987960054707 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_6$ (as 12T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 12
The 12 conjugacy class representatives for $C_6\times C_2$
Character table for $C_6\times C_2$

Intermediate fields

\(\Q(\sqrt{-259}) \), \(\Q(\sqrt{-111}) \), \(\Q(\sqrt{21}) \), \(\Q(\zeta_{7})^+\), \(\Q(\sqrt{21}, \sqrt{-111})\), 6.0.851324971.1, 6.0.3283682031.3, \(\Q(\zeta_{21})^+\)

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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/5.3.0.1}{3} }^{4}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{4}$

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
$3$3.12.6.2$x^{12} + 108 x^{6} - 243 x^{2} + 2916$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$7$7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
$37$37.6.3.1$x^{6} - 74 x^{4} + 1369 x^{2} - 202612$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
37.6.3.1$x^{6} - 74 x^{4} + 1369 x^{2} - 202612$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$