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Properties

Label 12.0.55956996935...3125.1
Degree $12$
Signature $[0, 6]$
Discriminant $5^{9}\cdot 7^{8}\cdot 89^{6}$
Root discriminant $115.43$
Ramified primes $5, 7, 89$
Class number $155780$ (GRH)
Class group $[2, 77890]$ (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![19212337691, 640091877, 2278832068, 52702688, 112725316, 1390204, 2950734, 8448, 42703, -101, 323, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 + 323*x^10 - 101*x^9 + 42703*x^8 + 8448*x^7 + 2950734*x^6 + 1390204*x^5 + 112725316*x^4 + 52702688*x^3 + 2278832068*x^2 + 640091877*x + 19212337691)
 
gp: K = bnfinit(x^12 - x^11 + 323*x^10 - 101*x^9 + 42703*x^8 + 8448*x^7 + 2950734*x^6 + 1390204*x^5 + 112725316*x^4 + 52702688*x^3 + 2278832068*x^2 + 640091877*x + 19212337691, 1)
 

Normalized defining polynomial

\( x^{12} - x^{11} + 323 x^{10} - 101 x^{9} + 42703 x^{8} + 8448 x^{7} + 2950734 x^{6} + 1390204 x^{5} + 112725316 x^{4} + 52702688 x^{3} + 2278832068 x^{2} + 640091877 x + 19212337691 \)

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:  \(5595699693580593283203125=5^{9}\cdot 7^{8}\cdot 89^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $115.43$
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:  $5, 7, 89$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3115=5\cdot 7\cdot 89\)
Dirichlet character group:    $\lbrace$$\chi_{3115}(2671,·)$, $\chi_{3115}(1,·)$, $\chi_{3115}(2402,·)$, $\chi_{3115}(1957,·)$, $\chi_{3115}(1514,·)$, $\chi_{3115}(1423,·)$, $\chi_{3115}(624,·)$, $\chi_{3115}(177,·)$, $\chi_{3115}(179,·)$, $\chi_{3115}(533,·)$, $\chi_{3115}(88,·)$, $\chi_{3115}(891,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{29} a^{8} + \frac{2}{29} a^{7} - \frac{9}{29} a^{6} + \frac{14}{29} a^{5} + \frac{4}{29} a^{4} + \frac{12}{29} a^{3} + \frac{6}{29} a^{2} - \frac{8}{29} a + \frac{1}{29}$, $\frac{1}{551} a^{9} + \frac{9}{551} a^{8} - \frac{169}{551} a^{7} + \frac{125}{551} a^{6} + \frac{15}{551} a^{5} + \frac{69}{551} a^{4} - \frac{258}{551} a^{3} - \frac{111}{551} a^{2} + \frac{61}{551} a + \frac{268}{551}$, $\frac{1}{551} a^{10} - \frac{3}{551} a^{8} - \frac{64}{551} a^{7} - \frac{27}{551} a^{6} + \frac{86}{551} a^{5} + \frac{109}{551} a^{4} + \frac{216}{551} a^{3} - \frac{213}{551} a^{2} - \frac{53}{551} a + \frac{39}{551}$, $\frac{1}{3201689199815754388060198533280005949} a^{11} + \frac{2530582730493434434254023836005315}{3201689199815754388060198533280005949} a^{10} + \frac{956869627632898417152365281037424}{3201689199815754388060198533280005949} a^{9} - \frac{1550528750238755678479044553973131}{3201689199815754388060198533280005949} a^{8} + \frac{11317990435061698413762532589072809}{110403075855715668553799949423448481} a^{7} - \frac{702184085483228386118994492213854466}{3201689199815754388060198533280005949} a^{6} - \frac{1542575330166452602065292613710756848}{3201689199815754388060198533280005949} a^{5} - \frac{618686005057921822653655951504861865}{3201689199815754388060198533280005949} a^{4} - \frac{1015326901720457780501635236180588250}{3201689199815754388060198533280005949} a^{3} - \frac{321604267088359633237643542110260387}{3201689199815754388060198533280005949} a^{2} - \frac{22710943409189125059358396976089115}{3201689199815754388060198533280005949} a - \frac{198812161080236419205845408510438508}{3201689199815754388060198533280005949}$

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

Class group and class number

$C_{2}\times C_{77890}$, which has order $155780$ (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

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{7})^+\), 4.0.990125.2, 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 ${\href{/LocalNumberField/11.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/17.12.0.1}{12} }$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/23.12.0.1}{12} }$ ${\href{/LocalNumberField/29.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/37.12.0.1}{12} }$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/47.12.0.1}{12} }$ ${\href{/LocalNumberField/53.12.0.1}{12} }$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{4}$

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}$
$89$89.6.3.2$x^{6} - 7921 x^{2} + 4934783$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
89.6.3.2$x^{6} - 7921 x^{2} + 4934783$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$