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Properties

Label 12.0.21571505642...8125.1
Degree $12$
Signature $[0, 6]$
Discriminant $3^{16}\cdot 5^{9}\cdot 37^{6}$
Root discriminant $88.00$
Ramified primes $3, 5, 37$
Class number $25480$ (GRH)
Class group $[2, 14, 910]$ (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![160801651, -36067335, 38602548, -6735844, 4054374, -535884, 228508, -20358, 7305, -381, 129, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 3*x^11 + 129*x^10 - 381*x^9 + 7305*x^8 - 20358*x^7 + 228508*x^6 - 535884*x^5 + 4054374*x^4 - 6735844*x^3 + 38602548*x^2 - 36067335*x + 160801651)
 
gp: K = bnfinit(x^12 - 3*x^11 + 129*x^10 - 381*x^9 + 7305*x^8 - 20358*x^7 + 228508*x^6 - 535884*x^5 + 4054374*x^4 - 6735844*x^3 + 38602548*x^2 - 36067335*x + 160801651, 1)
 

Normalized defining polynomial

\( x^{12} - 3 x^{11} + 129 x^{10} - 381 x^{9} + 7305 x^{8} - 20358 x^{7} + 228508 x^{6} - 535884 x^{5} + 4054374 x^{4} - 6735844 x^{3} + 38602548 x^{2} - 36067335 x + 160801651 \)

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:  \(215715056426865017578125=3^{16}\cdot 5^{9}\cdot 37^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $88.00$
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:  $3, 5, 37$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1665=3^{2}\cdot 5\cdot 37\)
Dirichlet character group:    $\lbrace$$\chi_{1665}(1,·)$, $\chi_{1665}(1444,·)$, $\chi_{1665}(517,·)$, $\chi_{1665}(73,·)$, $\chi_{1665}(556,·)$, $\chi_{1665}(334,·)$, $\chi_{1665}(1072,·)$, $\chi_{1665}(628,·)$, $\chi_{1665}(1111,·)$, $\chi_{1665}(889,·)$, $\chi_{1665}(1627,·)$, $\chi_{1665}(1183,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{19} a^{8} + \frac{1}{19} a^{7} - \frac{5}{19} a^{6} - \frac{5}{19} a^{5} + \frac{9}{19} a^{4} - \frac{3}{19} a^{3} + \frac{1}{19} a^{2} + \frac{7}{19} a - \frac{6}{19}$, $\frac{1}{2071} a^{9} - \frac{30}{2071} a^{8} + \frac{154}{2071} a^{7} - \frac{876}{2071} a^{6} - \frac{64}{2071} a^{5} - \frac{909}{2071} a^{4} - \frac{514}{2071} a^{3} + \frac{1002}{2071} a^{2} + \frac{138}{2071} a - \frac{555}{2071}$, $\frac{1}{2071} a^{10} + \frac{17}{2071} a^{8} + \frac{365}{2071} a^{7} + \frac{906}{2071} a^{6} - \frac{431}{2071} a^{5} - \frac{207}{2071} a^{4} - \frac{139}{2071} a^{3} - \frac{104}{2071} a^{2} + \frac{642}{2071} a - \frac{518}{2071}$, $\frac{1}{7261804429365008085368348321} a^{11} - \frac{57977464543590943609970}{382200233124474109756228859} a^{10} + \frac{946624993173542428174730}{7261804429365008085368348321} a^{9} - \frac{68689325147574016897171019}{7261804429365008085368348321} a^{8} - \frac{497268243336208516071279057}{7261804429365008085368348321} a^{7} + \frac{1472532204712477262421934312}{7261804429365008085368348321} a^{6} - \frac{3621974923884971172664130098}{7261804429365008085368348321} a^{5} + \frac{2788842572820432080244355698}{7261804429365008085368348321} a^{4} - \frac{2350485107428525020945452065}{7261804429365008085368348321} a^{3} - \frac{1283224198929347492880393577}{7261804429365008085368348321} a^{2} + \frac{162083869942721182042624506}{382200233124474109756228859} a + \frac{1638173877972836954649758777}{7261804429365008085368348321}$

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

Class group and class number

$C_{2}\times C_{14}\times C_{910}$, which has order $25480$ (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

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{9})^+\), 4.0.171125.1, 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 ${\href{/LocalNumberField/7.12.0.1}{12} }$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/13.12.0.1}{12} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/19.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/23.12.0.1}{12} }$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/43.12.0.1}{12} }$ ${\href{/LocalNumberField/47.12.0.1}{12} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{4}$

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
$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}$$[2]^{4}$
$5$5.12.9.1$x^{12} - 10 x^{8} - 375 x^{4} - 2000$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
$37$37.4.2.2$x^{4} - 37 x^{2} + 6845$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
37.4.2.2$x^{4} - 37 x^{2} + 6845$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
37.4.2.2$x^{4} - 37 x^{2} + 6845$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$