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Properties

Label 12.0.35263873843...0000.1
Degree $12$
Signature $[0, 6]$
Discriminant $2^{22}\cdot 3^{16}\cdot 5^{9}$
Root discriminant $51.56$
Ramified primes $2, 3, 5$
Class number $128$ (GRH)
Class group $[2, 8, 8]$ (GRH)
Galois group $C_3 : C_4$ (as 12T5)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![299120, 107040, 89040, 640, 12216, 960, 1296, -240, 216, 0, 6, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 + 6*x^10 + 216*x^8 - 240*x^7 + 1296*x^6 + 960*x^5 + 12216*x^4 + 640*x^3 + 89040*x^2 + 107040*x + 299120)
 
gp: K = bnfinit(x^12 + 6*x^10 + 216*x^8 - 240*x^7 + 1296*x^6 + 960*x^5 + 12216*x^4 + 640*x^3 + 89040*x^2 + 107040*x + 299120, 1)
 

Normalized defining polynomial

\( x^{12} + 6 x^{10} + 216 x^{8} - 240 x^{7} + 1296 x^{6} + 960 x^{5} + 12216 x^{4} + 640 x^{3} + 89040 x^{2} + 107040 x + 299120 \)

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:  \(352638738432000000000=2^{22}\cdot 3^{16}\cdot 5^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $51.56$
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:  $2, 3, 5$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{2} a^{4}$, $\frac{1}{2} a^{5}$, $\frac{1}{4} a^{6}$, $\frac{1}{4} a^{7}$, $\frac{1}{8} a^{8} - \frac{1}{2} a^{2}$, $\frac{1}{8} a^{9}$, $\frac{1}{85672} a^{10} - \frac{2423}{85672} a^{9} - \frac{433}{21418} a^{8} + \frac{2449}{42836} a^{7} + \frac{3071}{42836} a^{6} - \frac{1252}{10709} a^{5} + \frac{1006}{10709} a^{4} - \frac{1902}{10709} a^{3} + \frac{485}{10709} a^{2} - \frac{3430}{10709} a - \frac{1765}{10709}$, $\frac{1}{1457926765328308304} a^{11} - \frac{1991699705665}{364481691332077076} a^{10} - \frac{13339094353833597}{364481691332077076} a^{9} + \frac{9771379607961407}{182240845666038538} a^{8} + \frac{6998309401415507}{91120422833019269} a^{7} + \frac{10380645548989203}{91120422833019269} a^{6} + \frac{19125255475828787}{364481691332077076} a^{5} + \frac{16537342882710938}{91120422833019269} a^{4} + \frac{31909717174294965}{182240845666038538} a^{3} - \frac{20081227936090401}{91120422833019269} a^{2} + \frac{2922360124698434}{91120422833019269} a - \frac{28947528756344872}{91120422833019269}$

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

Class group and class number

$C_{2}\times C_{8}\times C_{8}$, which has order $128$ (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:  \( 1592.03444914 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3:C_4$ (as 12T5):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 12
The 6 conjugacy class representatives for $C_3 : C_4$
Character table for $C_3 : C_4$

Intermediate fields

\(\Q(\sqrt{5}) \), 3.3.1620.1 x3, 4.0.8000.2, 6.6.13122000.1

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 R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{4}$

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
$2$2.12.22.63$x^{12} - 60 x^{6} + 52$$6$$2$$22$$C_3 : C_4$$[3]_{3}^{2}$
$3$3.12.16.30$x^{12} + 93 x^{11} + 351 x^{10} + 3 x^{9} + 126 x^{8} - 297 x^{7} + 171 x^{6} + 243 x^{5} - 324 x^{4} - 54 x^{3} + 162 x^{2} - 243 x + 324$$3$$4$$16$$C_3 : C_4$$[2]^{4}$
$5$5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$