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Properties

Label 12.0.18883102336...8125.1
Degree $12$
Signature $[0, 6]$
Discriminant $5^{9}\cdot 7^{8}\cdot 109^{6}$
Root discriminant $127.74$
Ramified primes $5, 7, 109$
Class number $184820$ (GRH)
Class group $[2, 92410]$ (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![62174703641, 1786052977, 6155914968, 119660013, 252526341, 2549904, 5445884, 12398, 64553, -126, 398, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 + 398*x^10 - 126*x^9 + 64553*x^8 + 12398*x^7 + 5445884*x^6 + 2549904*x^5 + 252526341*x^4 + 119660013*x^3 + 6155914968*x^2 + 1786052977*x + 62174703641)
 
gp: K = bnfinit(x^12 - x^11 + 398*x^10 - 126*x^9 + 64553*x^8 + 12398*x^7 + 5445884*x^6 + 2549904*x^5 + 252526341*x^4 + 119660013*x^3 + 6155914968*x^2 + 1786052977*x + 62174703641, 1)
 

Normalized defining polynomial

\( x^{12} - x^{11} + 398 x^{10} - 126 x^{9} + 64553 x^{8} + 12398 x^{7} + 5445884 x^{6} + 2549904 x^{5} + 252526341 x^{4} + 119660013 x^{3} + 6155914968 x^{2} + 1786052977 x + 62174703641 \)

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:  \(18883102336086538361328125=5^{9}\cdot 7^{8}\cdot 109^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $127.74$
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:  $5, 7, 109$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3815=5\cdot 7\cdot 109\)
Dirichlet character group:    $\lbrace$$\chi_{3815}(2944,·)$, $\chi_{3815}(1,·)$, $\chi_{3815}(2181,·)$, $\chi_{3815}(3271,·)$, $\chi_{3815}(764,·)$, $\chi_{3815}(653,·)$, $\chi_{3815}(1198,·)$, $\chi_{3815}(3378,·)$, $\chi_{3815}(219,·)$, $\chi_{3815}(1852,·)$, $\chi_{3815}(2942,·)$, $\chi_{3815}(3487,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{811} a^{9} + \frac{81}{811} a^{8} - \frac{65}{811} a^{7} - \frac{203}{811} a^{6} - \frac{233}{811} a^{5} - \frac{327}{811} a^{4} - \frac{244}{811} a^{3} - \frac{246}{811} a^{2} + \frac{159}{811} a - \frac{150}{811}$, $\frac{1}{811} a^{10} - \frac{138}{811} a^{8} + \frac{196}{811} a^{7} - \frac{10}{811} a^{6} - \frac{107}{811} a^{5} + \frac{291}{811} a^{4} + \frac{54}{811} a^{3} - \frac{190}{811} a^{2} - \frac{53}{811} a - \frac{15}{811}$, $\frac{1}{3642964378009630247065324975126517347781} a^{11} + \frac{1974927219389753775002936736498604936}{3642964378009630247065324975126517347781} a^{10} + \frac{1240698990551064891031066695327854532}{3642964378009630247065324975126517347781} a^{9} + \frac{664006373424534505284947335895669507304}{3642964378009630247065324975126517347781} a^{8} + \frac{931509063521637415556996502235899147811}{3642964378009630247065324975126517347781} a^{7} - \frac{271714812777440308056529441169966812823}{3642964378009630247065324975126517347781} a^{6} + \frac{1075158425547598260918064982060057475957}{3642964378009630247065324975126517347781} a^{5} + \frac{1556983185724607979507018147968066289639}{3642964378009630247065324975126517347781} a^{4} + \frac{1332951689643241972827682567887233541615}{3642964378009630247065324975126517347781} a^{3} - \frac{1104269151565696950064019791354214273590}{3642964378009630247065324975126517347781} a^{2} - \frac{342079296660544642698133660571537173460}{3642964378009630247065324975126517347781} a + \frac{460841275118560041690116865028310771278}{3642964378009630247065324975126517347781}$

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

Class group and class number

$C_{2}\times C_{92410}$, which has order $184820$ (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

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{7})^+\), 4.0.1485125.1, 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 ${\href{/LocalNumberField/2.12.0.1}{12} }$ ${\href{/LocalNumberField/3.12.0.1}{12} }$ R R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/17.12.0.1}{12} }$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/23.12.0.1}{12} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/37.12.0.1}{12} }$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/47.12.0.1}{12} }$ ${\href{/LocalNumberField/53.12.0.1}{12} }$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{4}$

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
$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}$
$109$109.6.3.2$x^{6} - 11881 x^{2} + 12950290$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
109.6.3.2$x^{6} - 11881 x^{2} + 12950290$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$