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Properties

Label 12.0.33096636485...224.27
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 $4684800$ (GRH)
Class group $[2, 2, 2, 2, 40, 7320]$ (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![7114036000, -84951680, 287379984, -55312800, 14402624, -2403456, 842072, -73744, 10554, -760, 73, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 2*x^11 + 73*x^10 - 760*x^9 + 10554*x^8 - 73744*x^7 + 842072*x^6 - 2403456*x^5 + 14402624*x^4 - 55312800*x^3 + 287379984*x^2 - 84951680*x + 7114036000)
 
gp: K = bnfinit(x^12 - 2*x^11 + 73*x^10 - 760*x^9 + 10554*x^8 - 73744*x^7 + 842072*x^6 - 2403456*x^5 + 14402624*x^4 - 55312800*x^3 + 287379984*x^2 - 84951680*x + 7114036000, 1)
 

Normalized defining polynomial

\( x^{12} - 2 x^{11} + 73 x^{10} - 760 x^{9} + 10554 x^{8} - 73744 x^{7} + 842072 x^{6} - 2403456 x^{5} + 14402624 x^{4} - 55312800 x^{3} + 287379984 x^{2} - 84951680 x + 7114036000 \)

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$$\chi_{3192}(1,·)$, $\chi_{3192}(2507,·)$, $\chi_{3192}(2273,·)$, $\chi_{3192}(521,·)$, $\chi_{3192}(107,·)$, $\chi_{3192}(619,·)$, $\chi_{3192}(1873,·)$, $\chi_{3192}(179,·)$, $\chi_{3192}(691,·)$, $\chi_{3192}(2393,·)$, $\chi_{3192}(121,·)$, $\chi_{3192}(1483,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{20} a^{4} + \frac{3}{10} a^{3} + \frac{1}{20} a^{2} - \frac{1}{5} a - \frac{1}{2}$, $\frac{1}{20} a^{5} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{3}{10} a$, $\frac{1}{1120} a^{6} + \frac{3}{560} a^{5} - \frac{13}{1120} a^{4} + \frac{103}{280} a^{3} - \frac{61}{280} a^{2} - \frac{2}{35} a - \frac{15}{56}$, $\frac{1}{11200} a^{7} + \frac{1}{2800} a^{6} + \frac{31}{11200} a^{5} - \frac{117}{5600} a^{4} + \frac{151}{560} a^{3} + \frac{319}{1400} a^{2} - \frac{59}{560} a - \frac{25}{56}$, $\frac{1}{22400} a^{8} + \frac{1}{4480} a^{6} - \frac{209}{11200} a^{5} + \frac{83}{11200} a^{4} + \frac{477}{1400} a^{3} - \frac{417}{5600} a^{2} - \frac{159}{560} a - \frac{25}{112}$, $\frac{1}{224000} a^{9} - \frac{1}{112000} a^{8} + \frac{1}{44800} a^{7} - \frac{3}{14000} a^{6} - \frac{2279}{112000} a^{5} - \frac{1}{224} a^{4} + \frac{25667}{56000} a^{3} - \frac{647}{7000} a^{2} + \frac{1097}{5600} a - \frac{3}{14}$, $\frac{1}{250880000} a^{10} - \frac{27}{25088000} a^{9} + \frac{2971}{250880000} a^{8} - \frac{881}{31360000} a^{7} - \frac{127}{640000} a^{6} + \frac{129}{80000} a^{5} + \frac{11589}{640000} a^{4} + \frac{2181601}{7840000} a^{3} - \frac{2913563}{15680000} a^{2} - \frac{384429}{1568000} a + \frac{14759}{125440}$, $\frac{1}{593143710841830400000} a^{11} - \frac{19780025087}{296571855420915200000} a^{10} + \frac{1010236448269851}{593143710841830400000} a^{9} - \frac{1067773199998579}{74142963855228800000} a^{8} - \frac{2493836898015299}{74142963855228800000} a^{7} + \frac{113322087971237}{264796299482960000} a^{6} + \frac{129879307975618827}{10591851979318400000} a^{5} - \frac{76076539316444367}{3707148192761440000} a^{4} - \frac{7203141953185317171}{37071481927614400000} a^{3} - \frac{4765501641700582969}{18535740963807200000} a^{2} - \frac{3242262982494424993}{7414296385522880000} a + \frac{3811277174294059}{18535740963807200}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{40}\times C_{7320}$, which has order $4684800$ (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:  \( 452719.52039602736 \) (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

\(\Q(\sqrt{-114}) \), \(\Q(\sqrt{14}) \), \(\Q(\sqrt{-399}) \), 3.3.17689.2, \(\Q(\sqrt{14}, \sqrt{-114})\), 6.0.82185251774976.2, 6.6.1121436184064.1, 6.0.1123626489111.2

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 ${\href{/LocalNumberField/5.1.0.1}{1} }^{12}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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$2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.3$x^{2} + 2$$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}$