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Properties

Label 12.0.96878974764...0001.1
Degree $12$
Signature $[0, 6]$
Discriminant $3^{6}\cdot 7^{10}\cdot 19^{6}$
Root discriminant $38.21$
Ramified primes $3, 7, 19$
Class number $168$
Class group $[168]$
Galois group $C_6\times C_2$ (as 12T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![89971, -14525, 50834, -11802, 13720, -3780, 2493, -632, 312, -64, 23, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^11 + 23*x^10 - 64*x^9 + 312*x^8 - 632*x^7 + 2493*x^6 - 3780*x^5 + 13720*x^4 - 11802*x^3 + 50834*x^2 - 14525*x + 89971)
 
gp: K = bnfinit(x^12 - 4*x^11 + 23*x^10 - 64*x^9 + 312*x^8 - 632*x^7 + 2493*x^6 - 3780*x^5 + 13720*x^4 - 11802*x^3 + 50834*x^2 - 14525*x + 89971, 1)
 

Normalized defining polynomial

\( x^{12} - 4 x^{11} + 23 x^{10} - 64 x^{9} + 312 x^{8} - 632 x^{7} + 2493 x^{6} - 3780 x^{5} + 13720 x^{4} - 11802 x^{3} + 50834 x^{2} - 14525 x + 89971 \)

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:  \(9687897476476640001=3^{6}\cdot 7^{10}\cdot 19^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.21$
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:  $3, 7, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(399=3\cdot 7\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{399}(1,·)$, $\chi_{399}(227,·)$, $\chi_{399}(37,·)$, $\chi_{399}(362,·)$, $\chi_{399}(172,·)$, $\chi_{399}(398,·)$, $\chi_{399}(20,·)$, $\chi_{399}(341,·)$, $\chi_{399}(151,·)$, $\chi_{399}(248,·)$, $\chi_{399}(58,·)$, $\chi_{399}(379,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{587} a^{10} - \frac{91}{587} a^{9} + \frac{173}{587} a^{8} + \frac{196}{587} a^{7} + \frac{235}{587} a^{6} - \frac{186}{587} a^{5} + \frac{216}{587} a^{4} - \frac{211}{587} a^{3} - \frac{234}{587} a^{2} + \frac{271}{587} a - \frac{206}{587}$, $\frac{1}{704022234341983061} a^{11} + \frac{5192276514186}{24276628770413209} a^{10} - \frac{55403182339678400}{704022234341983061} a^{9} - \frac{186265927809104148}{704022234341983061} a^{8} - \frac{93789680828306301}{704022234341983061} a^{7} - \frac{23790922603745059}{704022234341983061} a^{6} + \frac{15782900066731261}{704022234341983061} a^{5} + \frac{152609719293345315}{704022234341983061} a^{4} + \frac{121313606388379427}{704022234341983061} a^{3} + \frac{332076716660729618}{704022234341983061} a^{2} + \frac{9418295269481208}{24276628770413209} a + \frac{205325555860862019}{704022234341983061}$

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

Class group and class number

$C_{168}$, which has order $168$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 140.798796005 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_6$ (as 12T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 12
The 12 conjugacy class representatives for $C_6\times C_2$
Character table for $C_6\times C_2$

Intermediate fields

\(\Q(\sqrt{21}) \), \(\Q(\sqrt{-19}) \), \(\Q(\sqrt{-399}) \), \(\Q(\zeta_{7})^+\), \(\Q(\sqrt{-19}, \sqrt{21})\), \(\Q(\zeta_{21})^+\), 6.0.16468459.1, 6.0.3112538751.2

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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/5.3.0.1}{3} }^{4}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{4}$ R ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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$3.12.6.2$x^{12} + 108 x^{6} - 243 x^{2} + 2916$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$7$7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
$19$19.12.6.1$x^{12} + 41154 x^{6} - 2476099 x^{2} + 423412929$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$