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Properties

Label 12.0.37176212245...3808.1
Degree $12$
Signature $[0, 6]$
Discriminant $2^{12}\cdot 3^{16}\cdot 7^{6}\cdot 13^{11}$
Root discriminant $240.36$
Ramified primes $2, 3, 7, 13$
Class number $1286064$ (GRH)
Class group $[2, 6, 6, 17862]$ (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![3678675897, -807602328, 427391367, 12075388, 4214691, 204828, 942422, 25428, 20787, -52, 195, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 + 195*x^10 - 52*x^9 + 20787*x^8 + 25428*x^7 + 942422*x^6 + 204828*x^5 + 4214691*x^4 + 12075388*x^3 + 427391367*x^2 - 807602328*x + 3678675897)
 
gp: K = bnfinit(x^12 + 195*x^10 - 52*x^9 + 20787*x^8 + 25428*x^7 + 942422*x^6 + 204828*x^5 + 4214691*x^4 + 12075388*x^3 + 427391367*x^2 - 807602328*x + 3678675897, 1)
 

Normalized defining polynomial

\( x^{12} + 195 x^{10} - 52 x^{9} + 20787 x^{8} + 25428 x^{7} + 942422 x^{6} + 204828 x^{5} + 4214691 x^{4} + 12075388 x^{3} + 427391367 x^{2} - 807602328 x + 3678675897 \)

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:  \(37176212245675331887703543808=2^{12}\cdot 3^{16}\cdot 7^{6}\cdot 13^{11}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $240.36$
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:  $2, 3, 7, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3276=2^{2}\cdot 3^{2}\cdot 7\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{3276}(1,·)$, $\chi_{3276}(2659,·)$, $\chi_{3276}(673,·)$, $\chi_{3276}(841,·)$, $\chi_{3276}(811,·)$, $\chi_{3276}(589,·)$, $\chi_{3276}(307,·)$, $\chi_{3276}(643,·)$, $\chi_{3276}(2521,·)$, $\chi_{3276}(2941,·)$, $\chi_{3276}(1987,·)$, $\chi_{3276}(223,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{26} a^{6} - \frac{1}{2}$, $\frac{1}{26} a^{7} - \frac{1}{2} a$, $\frac{1}{312} a^{8} + \frac{1}{156} a^{7} + \frac{1}{312} a^{6} + \frac{1}{3} a^{5} + \frac{1}{6} a^{4} + \frac{1}{3} a^{3} - \frac{1}{8} a^{2} + \frac{1}{4} a + \frac{3}{8}$, $\frac{1}{312} a^{9} - \frac{1}{104} a^{7} - \frac{1}{52} a^{6} - \frac{1}{2} a^{5} + \frac{5}{24} a^{3} - \frac{1}{2} a^{2} - \frac{1}{8} a - \frac{1}{4}$, $\frac{1}{18096} a^{10} - \frac{1}{696} a^{9} + \frac{7}{4524} a^{8} + \frac{1}{696} a^{7} + \frac{103}{18096} a^{6} - \frac{25}{87} a^{5} + \frac{227}{464} a^{4} - \frac{343}{696} a^{3} - \frac{21}{58} a^{2} - \frac{3}{232} a - \frac{3}{16}$, $\frac{1}{8255524250010437208804802433641872624} a^{11} + \frac{12642475649589197517897426505705}{8255524250010437208804802433641872624} a^{10} - \frac{3499080647573241775068451770254749}{4127762125005218604402401216820936312} a^{9} + \frac{79343883773699297270905048003561}{687960354167536434067066869470156052} a^{8} + \frac{8820442206176627704530169289314291}{2751841416670145736268267477880624208} a^{7} + \frac{46245469286562474275700830786771689}{2751841416670145736268267477880624208} a^{6} + \frac{308410580382165979656390456275245001}{635040326923879785292677110280144048} a^{5} + \frac{26889522878205771499804558770369317}{635040326923879785292677110280144048} a^{4} + \frac{104558712280671310858819384963704835}{317520163461939892646338555140072024} a^{3} - \frac{6804365276438996655706409986384985}{17640009081218882924796586396670668} a^{2} + \frac{1352753985260792969800525584589691}{211680108974626595097559036760048016} a - \frac{342323219998240305303677719865279}{7299314102573330865433070233105104}$

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

Class group and class number

$C_{2}\times C_{6}\times C_{6}\times C_{17862}$, which has order $1286064$ (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:  \( 16557.868316895623 \) (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

\(\Q(\sqrt{13}) \), 3.3.13689.1, 4.0.1722448.1, 6.6.2436053373.1

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 R R ${\href{/LocalNumberField/5.12.0.1}{12} }$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}$ R ${\href{/LocalNumberField/17.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/19.12.0.1}{12} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/29.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/31.12.0.1}{12} }$ ${\href{/LocalNumberField/37.12.0.1}{12} }$ ${\href{/LocalNumberField/41.12.0.1}{12} }$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/47.12.0.1}{12} }$ ${\href{/LocalNumberField/53.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}$

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
$2$2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
$3$3.3.4.3$x^{3} - 3 x^{2} + 12$$3$$1$$4$$C_3$$[2]$
3.3.4.3$x^{3} - 3 x^{2} + 12$$3$$1$$4$$C_3$$[2]$
3.3.4.3$x^{3} - 3 x^{2} + 12$$3$$1$$4$$C_3$$[2]$
3.3.4.3$x^{3} - 3 x^{2} + 12$$3$$1$$4$$C_3$$[2]$
$7$7.12.6.2$x^{12} + 7203 x^{4} - 16807 x^{2} + 588245$$2$$6$$6$$C_{12}$$[\ ]_{2}^{6}$
$13$13.12.11.3$x^{12} - 208$$12$$1$$11$$C_{12}$$[\ ]_{12}$