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Properties

Label 12.0.18694897744...8064.6
Degree $12$
Signature $[0, 6]$
Discriminant $2^{12}\cdot 3^{6}\cdot 7^{10}\cdot 53^{6}$
Root discriminant $127.64$
Ramified primes $2, 3, 7, 53$
Class number $170352$ (GRH)
Class group $[26, 6552]$ (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![28128047641, -1041821864, 2848609153, -86207266, 123571099, -2980318, 2941140, -53740, 40532, -506, 307, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 2*x^11 + 307*x^10 - 506*x^9 + 40532*x^8 - 53740*x^7 + 2941140*x^6 - 2980318*x^5 + 123571099*x^4 - 86207266*x^3 + 2848609153*x^2 - 1041821864*x + 28128047641)
 
gp: K = bnfinit(x^12 - 2*x^11 + 307*x^10 - 506*x^9 + 40532*x^8 - 53740*x^7 + 2941140*x^6 - 2980318*x^5 + 123571099*x^4 - 86207266*x^3 + 2848609153*x^2 - 1041821864*x + 28128047641, 1)
 

Normalized defining polynomial

\( x^{12} - 2 x^{11} + 307 x^{10} - 506 x^{9} + 40532 x^{8} - 53740 x^{7} + 2941140 x^{6} - 2980318 x^{5} + 123571099 x^{4} - 86207266 x^{3} + 2848609153 x^{2} - 1041821864 x + 28128047641 \)

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:  \(18694897744381964174168064=2^{12}\cdot 3^{6}\cdot 7^{10}\cdot 53^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $127.64$
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:  $2, 3, 7, 53$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4452=2^{2}\cdot 3\cdot 7\cdot 53\)
Dirichlet character group:    $\lbrace$$\chi_{4452}(1,·)$, $\chi_{4452}(2755,·)$, $\chi_{4452}(211,·)$, $\chi_{4452}(1697,·)$, $\chi_{4452}(4241,·)$, $\chi_{4452}(425,·)$, $\chi_{4452}(1907,·)$, $\chi_{4452}(635,·)$, $\chi_{4452}(2545,·)$, $\chi_{4452}(4451,·)$, $\chi_{4452}(3817,·)$, $\chi_{4452}(4027,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{112017608613902180664418416498954377} a^{11} + \frac{40491741564283713851616493394343615}{112017608613902180664418416498954377} a^{10} - \frac{2168400196396595181268287361322148}{112017608613902180664418416498954377} a^{9} - \frac{21364082651895707652565114753459257}{112017608613902180664418416498954377} a^{8} - \frac{37452115828330828082979443379110568}{112017608613902180664418416498954377} a^{7} - \frac{23888034756200427319575355975806097}{112017608613902180664418416498954377} a^{6} - \frac{590053097276534580275454636444512}{112017608613902180664418416498954377} a^{5} + \frac{36342089934196470332004743171938110}{112017608613902180664418416498954377} a^{4} - \frac{51293278740547862293224979466380301}{112017608613902180664418416498954377} a^{3} + \frac{17425453383420077674267115387185237}{112017608613902180664418416498954377} a^{2} - \frac{50520554417918889640364080725171110}{112017608613902180664418416498954377} a - \frac{12509893870215744820391555634156497}{112017608613902180664418416498954377}$

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

Class group and class number

$C_{26}\times C_{6552}$, which has order $170352$ (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

\(\Q(\sqrt{-1113}) \), \(\Q(\sqrt{21}) \), \(\Q(\sqrt{-53}) \), \(\Q(\zeta_{7})^+\), \(\Q(\sqrt{21}, \sqrt{-53})\), 6.0.4323759676992.3, \(\Q(\zeta_{21})^+\), 6.0.22877035328.4

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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
$2$2.12.12.26$x^{12} - 162 x^{10} + 26423 x^{8} + 125508 x^{6} - 64481 x^{4} - 122498 x^{2} - 86071$$2$$6$$12$$C_6\times C_2$$[2]^{6}$
$3$3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$7$7.12.10.1$x^{12} - 70 x^{6} + 35721$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$
$53$53.12.6.1$x^{12} + 2382032 x^{6} - 418195493 x^{2} + 1418519112256$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$