↑

Properties

Label 12.0.33096636485...224.23
Degree $12$
Signature $[0, 6]$
Discriminant $2^{18}\cdot 3^{6}\cdot 7^{10}\cdot 19^{10}$
Root discriminant $288.39$
Ramified primes $2, 3, 7, 19$
Class number $1218048$ (GRH)
Class group $[2, 2, 2, 2, 2, 38064]$ (GRH)
Galois group $C_6\times C_2$ (as 12T2)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2287368448, -129311488, 36508032, -27044448, 8722892, 501456, -73396, -57920, 11031, 760, -164, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^11 - 164*x^10 + 760*x^9 + 11031*x^8 - 57920*x^7 - 73396*x^6 + 501456*x^5 + 8722892*x^4 - 27044448*x^3 + 36508032*x^2 - 129311488*x + 2287368448)
 
gp: K = bnfinit(x^12 - 4*x^11 - 164*x^10 + 760*x^9 + 11031*x^8 - 57920*x^7 - 73396*x^6 + 501456*x^5 + 8722892*x^4 - 27044448*x^3 + 36508032*x^2 - 129311488*x + 2287368448, 1)
 

Normalized defining polynomial

\( x^{12} - 4 x^{11} - 164 x^{10} + 760 x^{9} + 11031 x^{8} - 57920 x^{7} - 73396 x^{6} + 501456 x^{5} + 8722892 x^{4} - 27044448 x^{3} + 36508032 x^{2} - 129311488 x + 2287368448 \)

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:  \(330966364856493592168450228224=2^{18}\cdot 3^{6}\cdot 7^{10}\cdot 19^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $288.39$
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:  $2, 3, 7, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3192=2^{3}\cdot 3\cdot 7\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{3192}(1,·)$, $\chi_{3192}(2819,·)$, $\chi_{3192}(1171,·)$, $\chi_{3192}(2273,·)$, $\chi_{3192}(379,·)$, $\chi_{3192}(521,·)$, $\chi_{3192}(1873,·)$, $\chi_{3192}(419,·)$, $\chi_{3192}(1243,·)$, $\chi_{3192}(2393,·)$, $\chi_{3192}(121,·)$, $\chi_{3192}(2747,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{532} a^{6} + \frac{131}{532} a^{5} + \frac{7}{76} a^{4} - \frac{187}{532} a^{3} + \frac{3}{19} a^{2} - \frac{39}{266} a + \frac{33}{133}$, $\frac{1}{532} a^{7} - \frac{22}{133} a^{5} + \frac{11}{133} a^{4} + \frac{109}{532} a^{3} - \frac{44}{133} a^{2} + \frac{121}{266} a + \frac{66}{133}$, $\frac{1}{2128} a^{8} - \frac{1}{1064} a^{6} + \frac{101}{532} a^{5} - \frac{465}{2128} a^{4} - \frac{2}{133} a^{3} - \frac{257}{1064} a^{2} - \frac{37}{133} a - \frac{22}{133}$, $\frac{1}{21280} a^{9} + \frac{1}{10640} a^{8} + \frac{1}{10640} a^{7} - \frac{1}{5320} a^{6} - \frac{5053}{21280} a^{5} - \frac{2311}{10640} a^{4} - \frac{2197}{10640} a^{3} + \frac{187}{760} a^{2} + \frac{41}{266} a - \frac{37}{95}$, $\frac{1}{56565602240} a^{10} + \frac{29321}{3535350140} a^{9} + \frac{94497}{707070028} a^{8} - \frac{258124}{883837535} a^{7} + \frac{37141551}{56565602240} a^{6} + \frac{270914759}{14141400560} a^{5} - \frac{50467331}{372142120} a^{4} + \frac{14282875}{1414140056} a^{3} + \frac{551804781}{14141400560} a^{2} + \frac{214791718}{883837535} a - \frac{1109567}{21297290}$, $\frac{1}{2799214409118768260480} a^{11} + \frac{425629643}{87475450284961508140} a^{10} - \frac{964250903424471}{99971943182813152160} a^{9} + \frac{23577065111205313}{174950900569923016280} a^{8} - \frac{3009420427344711}{29465414832829139584} a^{7} + \frac{89040413138175221}{99971943182813152160} a^{6} - \frac{156020973345404066097}{699803602279692065120} a^{5} - \frac{1736659049861261439}{49985971591406576080} a^{4} - \frac{326050580991494427521}{699803602279692065120} a^{3} - \frac{6945487002487109}{17178996521005795} a^{2} - \frac{22502200979850651}{87475450284961508140} a - \frac{14836663203434147}{37640038848950735}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{38064}$, which has order $1218048$ (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:  \( 208587.07105926925 \) (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

\(\Q(\sqrt{38}) \), \(\Q(\sqrt{-399}) \), \(\Q(\sqrt{-42}) \), 3.3.17689.2, \(\Q(\sqrt{38}, \sqrt{-42})\), 6.6.3043898213888.1, 6.0.1123626489111.2, 6.0.30278776969728.2

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 R R ${\href{/LocalNumberField/5.2.0.1}{2} }^{6}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{4}$ R ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/53.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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
$2$2.2.3.4$x^{2} + 10$$2$$1$$3$$C_2$$[3]$
2.2.3.4$x^{2} + 10$$2$$1$$3$$C_2$$[3]$
2.2.3.4$x^{2} + 10$$2$$1$$3$$C_2$$[3]$
2.2.3.4$x^{2} + 10$$2$$1$$3$$C_2$$[3]$
2.2.3.4$x^{2} + 10$$2$$1$$3$$C_2$$[3]$
2.2.3.4$x^{2} + 10$$2$$1$$3$$C_2$$[3]$
$3$3.12.6.2$x^{12} + 108 x^{6} - 243 x^{2} + 2916$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$7$7.12.10.1$x^{12} - 70 x^{6} + 35721$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$
$19$19.12.10.3$x^{12} - 19 x^{6} + 5776$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$