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Properties

Label 12.0.50010143639...8125.1
Degree $12$
Signature $[0, 6]$
Discriminant $3^{16}\cdot 5^{9}\cdot 29^{6}$
Root discriminant $77.91$
Ramified primes $3, 5, 29$
Class number $10756$ (GRH)
Class group $[2, 5378]$ (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![44340281, -11707065, 12629298, -2617004, 1615344, -258864, 111738, -12288, 4425, -291, 99, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 3*x^11 + 99*x^10 - 291*x^9 + 4425*x^8 - 12288*x^7 + 111738*x^6 - 258864*x^5 + 1615344*x^4 - 2617004*x^3 + 12629298*x^2 - 11707065*x + 44340281)
 
gp: K = bnfinit(x^12 - 3*x^11 + 99*x^10 - 291*x^9 + 4425*x^8 - 12288*x^7 + 111738*x^6 - 258864*x^5 + 1615344*x^4 - 2617004*x^3 + 12629298*x^2 - 11707065*x + 44340281, 1)
 

Normalized defining polynomial

\( x^{12} - 3 x^{11} + 99 x^{10} - 291 x^{9} + 4425 x^{8} - 12288 x^{7} + 111738 x^{6} - 258864 x^{5} + 1615344 x^{4} - 2617004 x^{3} + 12629298 x^{2} - 11707065 x + 44340281 \)

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:  \(50010143639414923828125=3^{16}\cdot 5^{9}\cdot 29^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $77.91$
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:  $3, 5, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1305=3^{2}\cdot 5\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{1305}(1,·)$, $\chi_{1305}(898,·)$, $\chi_{1305}(1219,·)$, $\chi_{1305}(871,·)$, $\chi_{1305}(202,·)$, $\chi_{1305}(463,·)$, $\chi_{1305}(1072,·)$, $\chi_{1305}(436,·)$, $\chi_{1305}(28,·)$, $\chi_{1305}(349,·)$, $\chi_{1305}(784,·)$, $\chi_{1305}(637,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{71} a^{9} - \frac{24}{71} a^{8} + \frac{33}{71} a^{7} + \frac{19}{71} a^{6} - \frac{3}{71} a^{5} + \frac{11}{71} a^{4} + \frac{12}{71} a^{3} - \frac{13}{71} a^{2} - \frac{15}{71} a$, $\frac{1}{71} a^{10} + \frac{25}{71} a^{8} + \frac{30}{71} a^{7} + \frac{27}{71} a^{6} + \frac{10}{71} a^{5} - \frac{8}{71} a^{4} - \frac{9}{71} a^{3} + \frac{28}{71} a^{2} - \frac{5}{71} a$, $\frac{1}{1026174231742957118179038624061} a^{11} - \frac{6264205092319832767792595910}{1026174231742957118179038624061} a^{10} + \frac{4290054201436971420796085433}{1026174231742957118179038624061} a^{9} - \frac{212720481912225804563241987017}{1026174231742957118179038624061} a^{8} + \frac{282852253126721962842755213433}{1026174231742957118179038624061} a^{7} - \frac{209912200438953798073593768498}{1026174231742957118179038624061} a^{6} + \frac{312487655216926307520422447811}{1026174231742957118179038624061} a^{5} + \frac{64004133698074365223599597636}{1026174231742957118179038624061} a^{4} + \frac{482776272952299272000319550}{2858424043852248240053032379} a^{3} - \frac{313390209227616215333303967978}{1026174231742957118179038624061} a^{2} - \frac{203570266572875713151884928323}{1026174231742957118179038624061} a - \frac{84533905521769782368571468}{760692536503304016441096089}$

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

Class group and class number

$C_{2}\times C_{5378}$, which has order $10756$ (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:  \( 201.000834787 \) (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

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{9})^+\), 4.0.105125.2, 6.6.820125.1

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 ${\href{/LocalNumberField/2.12.0.1}{12} }$ R R ${\href{/LocalNumberField/7.12.0.1}{12} }$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/13.12.0.1}{12} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/19.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/23.12.0.1}{12} }$ R ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/43.12.0.1}{12} }$ ${\href{/LocalNumberField/47.12.0.1}{12} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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
$3$3.12.16.14$x^{12} + 72 x^{11} - 36 x^{10} + 108 x^{9} - 108 x^{8} + 54 x^{7} + 72 x^{6} - 81 x^{5} - 81 x^{4} - 81 x^{3} + 81 x^{2} - 81$$3$$4$$16$$C_{12}$$[2]^{4}$
$5$5.12.9.2$x^{12} - 10 x^{8} + 25 x^{4} - 500$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
$29$29.6.3.2$x^{6} - 841 x^{2} + 73167$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
29.6.3.2$x^{6} - 841 x^{2} + 73167$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$