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Properties

Label 12.0.76595392638...000.16
Degree $12$
Signature $[0, 6]$
Discriminant $2^{18}\cdot 3^{18}\cdot 5^{6}\cdot 13^{6}$
Root discriminant $118.49$
Ramified primes $2, 3, 5, 13$
Class number $183008$ (GRH)
Class group $[4, 45752]$ (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![3841091281, -698770170, 753351309, -111141586, 60554802, -6081186, 2195273, -146022, 38814, -1550, 321, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 6*x^11 + 321*x^10 - 1550*x^9 + 38814*x^8 - 146022*x^7 + 2195273*x^6 - 6081186*x^5 + 60554802*x^4 - 111141586*x^3 + 753351309*x^2 - 698770170*x + 3841091281)
 
gp: K = bnfinit(x^12 - 6*x^11 + 321*x^10 - 1550*x^9 + 38814*x^8 - 146022*x^7 + 2195273*x^6 - 6081186*x^5 + 60554802*x^4 - 111141586*x^3 + 753351309*x^2 - 698770170*x + 3841091281, 1)
 

Normalized defining polynomial

\( x^{12} - 6 x^{11} + 321 x^{10} - 1550 x^{9} + 38814 x^{8} - 146022 x^{7} + 2195273 x^{6} - 6081186 x^{5} + 60554802 x^{4} - 111141586 x^{3} + 753351309 x^{2} - 698770170 x + 3841091281 \)

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:  \(7659539263855005696000000=2^{18}\cdot 3^{18}\cdot 5^{6}\cdot 13^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $118.49$
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:  $2, 3, 5, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4680=2^{3}\cdot 3^{2}\cdot 5\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{4680}(1,·)$, $\chi_{4680}(389,·)$, $\chi_{4680}(1249,·)$, $\chi_{4680}(3821,·)$, $\chi_{4680}(1949,·)$, $\chi_{4680}(4369,·)$, $\chi_{4680}(3121,·)$, $\chi_{4680}(2261,·)$, $\chi_{4680}(1561,·)$, $\chi_{4680}(2809,·)$, $\chi_{4680}(701,·)$, $\chi_{4680}(3509,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{26} a^{4} - \frac{1}{13} a^{3} - \frac{1}{26} a^{2} + \frac{1}{13} a + \frac{1}{26}$, $\frac{1}{26} a^{5} - \frac{5}{26} a^{3} + \frac{5}{26} a + \frac{1}{13}$, $\frac{1}{26} a^{6} - \frac{5}{13} a^{3} + \frac{6}{13} a + \frac{5}{26}$, $\frac{1}{26} a^{7} + \frac{3}{13} a^{3} + \frac{1}{13} a^{2} - \frac{1}{26} a + \frac{5}{13}$, $\frac{1}{676} a^{8} - \frac{1}{169} a^{7} + \frac{1}{338} a^{6} + \frac{2}{169} a^{5} - \frac{5}{676} a^{4} - \frac{2}{169} a^{3} + \frac{1}{338} a^{2} + \frac{1}{169} a + \frac{1}{676}$, $\frac{1}{676} a^{9} + \frac{3}{169} a^{7} - \frac{5}{338} a^{6} + \frac{1}{676} a^{5} - \frac{1}{338} a^{4} - \frac{53}{169} a^{3} + \frac{19}{338} a^{2} + \frac{277}{676} a + \frac{27}{169}$, $\frac{1}{44795252324836} a^{10} - \frac{5}{44795252324836} a^{9} - \frac{14974839225}{44795252324836} a^{8} + \frac{29949678465}{22397626162418} a^{7} - \frac{725603188975}{44795252324836} a^{6} + \frac{244267497463}{44795252324836} a^{5} + \frac{504065752465}{44795252324836} a^{4} + \frac{1099404912420}{11198813081209} a^{3} + \frac{21630443659391}{44795252324836} a^{2} + \frac{16976640116925}{44795252324836} a - \frac{42379329215}{99766708964}$, $\frac{1}{173672850687725190932} a^{11} + \frac{1938513}{173672850687725190932} a^{10} + \frac{430011927512241}{6679725026450968882} a^{9} + \frac{24557926036967747}{86836425343862595466} a^{8} - \frac{769081207120729313}{173672850687725190932} a^{7} + \frac{384906566370855831}{173672850687725190932} a^{6} - \frac{1630774464566484139}{86836425343862595466} a^{5} + \frac{819497330132987702}{43418212671931297733} a^{4} + \frac{77685517689378529561}{173672850687725190932} a^{3} - \frac{50207436183310573641}{173672850687725190932} a^{2} + \frac{1498561410693355279}{43418212671931297733} a + \frac{29861875262126657}{193399611010829834}$

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

Class group and class number

$C_{4}\times C_{45752}$, which has order $183008$ (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:  \( 201.0008347866989 \) (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

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{-390}) \), \(\Q(\sqrt{-78}) \), \(\Q(\zeta_{9})^+\), \(\Q(\sqrt{5}, \sqrt{-78})\), 6.6.820125.1, 6.0.2767587264000.2, 6.0.22140698112.9

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 R ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/19.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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$2.12.18.23$x^{12} + 52 x^{10} - 28 x^{8} + 8 x^{6} + 64 x^{4} - 32 x^{2} + 64$$2$$6$$18$$C_6\times C_2$$[3]^{6}$
$3$3.12.18.82$x^{12} - 9 x^{9} + 9 x^{8} - 9 x^{5} - 9 x^{4} - 9 x^{3} + 9$$6$$2$$18$$C_6\times C_2$$[2]_{2}^{2}$
$5$5.12.6.1$x^{12} + 500 x^{6} - 3125 x^{2} + 62500$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$13$13.12.6.1$x^{12} + 338 x^{8} + 8788 x^{6} + 28561 x^{4} + 19307236$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$