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Properties

Label 12.0.21696811346...0000.1
Degree $12$
Signature $[0, 6]$
Discriminant $2^{12}\cdot 5^{9}\cdot 7^{8}\cdot 19^{6}$
Root discriminant $106.67$
Ramified primes $2, 5, 7, 19$
Class number $100048$ (GRH)
Class group $[2, 2, 25012]$ (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![7573971901, -709042578, 1065197590, -81302524, 62575025, -3748466, 1935477, -84412, 32574, -926, 283, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^11 + 283*x^10 - 926*x^9 + 32574*x^8 - 84412*x^7 + 1935477*x^6 - 3748466*x^5 + 62575025*x^4 - 81302524*x^3 + 1065197590*x^2 - 709042578*x + 7573971901)
 
gp: K = bnfinit(x^12 - 4*x^11 + 283*x^10 - 926*x^9 + 32574*x^8 - 84412*x^7 + 1935477*x^6 - 3748466*x^5 + 62575025*x^4 - 81302524*x^3 + 1065197590*x^2 - 709042578*x + 7573971901, 1)
 

Normalized defining polynomial

\( x^{12} - 4 x^{11} + 283 x^{10} - 926 x^{9} + 32574 x^{8} - 84412 x^{7} + 1935477 x^{6} - 3748466 x^{5} + 62575025 x^{4} - 81302524 x^{3} + 1065197590 x^{2} - 709042578 x + 7573971901 \)

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:  \(2169681134677448000000000=2^{12}\cdot 5^{9}\cdot 7^{8}\cdot 19^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $106.67$
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:  $2, 5, 7, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2660=2^{2}\cdot 5\cdot 7\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{2660}(1,·)$, $\chi_{2660}(1443,·)$, $\chi_{2660}(2129,·)$, $\chi_{2660}(683,·)$, $\chi_{2660}(1901,·)$, $\chi_{2660}(303,·)$, $\chi_{2660}(1521,·)$, $\chi_{2660}(1747,·)$, $\chi_{2660}(1367,·)$, $\chi_{2660}(1369,·)$, $\chi_{2660}(989,·)$, $\chi_{2660}(2507,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{19} a^{6} - \frac{2}{19} a^{5} - \frac{3}{19} a^{4} + \frac{6}{19} a^{3} + \frac{2}{19} a^{2} - \frac{4}{19} a + \frac{1}{19}$, $\frac{1}{19} a^{7} - \frac{7}{19} a^{5} - \frac{5}{19} a^{3} - \frac{7}{19} a + \frac{2}{19}$, $\frac{1}{19} a^{8} + \frac{5}{19} a^{5} - \frac{7}{19} a^{4} + \frac{4}{19} a^{3} + \frac{7}{19} a^{2} - \frac{7}{19} a + \frac{7}{19}$, $\frac{1}{19} a^{9} + \frac{3}{19} a^{5} - \frac{4}{19} a^{3} + \frac{2}{19} a^{2} + \frac{8}{19} a - \frac{5}{19}$, $\frac{1}{353537829059} a^{10} + \frac{5707499532}{353537829059} a^{9} - \frac{3587687085}{353537829059} a^{8} + \frac{695522678}{353537829059} a^{7} - \frac{6072870191}{353537829059} a^{6} + \frac{154379928530}{353537829059} a^{5} + \frac{119820005127}{353537829059} a^{4} - \frac{15830530199}{353537829059} a^{3} + \frac{172234599430}{353537829059} a^{2} + \frac{82183816415}{353537829059} a - \frac{2015447223}{18607254161}$, $\frac{1}{10188774541094629674139559} a^{11} - \frac{626896787474}{10188774541094629674139559} a^{10} + \frac{34753355723068176291204}{10188774541094629674139559} a^{9} - \frac{79801988682279649509520}{10188774541094629674139559} a^{8} - \frac{53615051977602558090071}{10188774541094629674139559} a^{7} + \frac{90625568017827390015448}{10188774541094629674139559} a^{6} - \frac{23512441571045528714796}{536251291636559456533661} a^{5} - \frac{3296657991001717769656515}{10188774541094629674139559} a^{4} - \frac{3132890309848291848606046}{10188774541094629674139559} a^{3} - \frac{4470892829520868639395614}{10188774541094629674139559} a^{2} + \frac{2850997497215183186137996}{10188774541094629674139559} a - \frac{1965742524934995733181193}{10188774541094629674139559}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{25012}$, which has order $100048$ (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:  \( 104.882003477 \) (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

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{7})^+\), 4.0.722000.3, 6.6.300125.1

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 ${\href{/LocalNumberField/3.12.0.1}{12} }$ R R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/17.12.0.1}{12} }$ R ${\href{/LocalNumberField/23.12.0.1}{12} }$ ${\href{/LocalNumberField/29.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/37.12.0.1}{12} }$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/47.12.0.1}{12} }$ ${\href{/LocalNumberField/53.12.0.1}{12} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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$2.12.12.25$x^{12} - 78 x^{10} - 1621 x^{8} + 460 x^{6} - 1977 x^{4} + 866 x^{2} + 749$$2$$6$$12$$C_{12}$$[2]^{6}$
$5$5.12.9.2$x^{12} - 10 x^{8} + 25 x^{4} - 500$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
$7$7.12.8.1$x^{12} - 63 x^{9} + 637 x^{6} + 6174 x^{3} + 300125$$3$$4$$8$$C_{12}$$[\ ]_{3}^{4}$
$19$19.6.3.2$x^{6} - 361 x^{2} + 27436$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
19.6.3.2$x^{6} - 361 x^{2} + 27436$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$