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Properties

Label 12.0.33511670005...8329.4
Degree $12$
Signature $[0, 6]$
Discriminant $13^{10}\cdot 79^{6}$
Root discriminant $75.35$
Ramified primes $13, 79$
Class number $10080$ (GRH)
Class group $[2, 2, 2, 6, 210]$ (GRH)
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![101631179, -15437955, 25059307, -3326843, 2706834, -302185, 163344, -14486, 5800, -368, 115, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^11 + 115*x^10 - 368*x^9 + 5800*x^8 - 14486*x^7 + 163344*x^6 - 302185*x^5 + 2706834*x^4 - 3326843*x^3 + 25059307*x^2 - 15437955*x + 101631179)
 
gp: K = bnfinit(x^12 - 4*x^11 + 115*x^10 - 368*x^9 + 5800*x^8 - 14486*x^7 + 163344*x^6 - 302185*x^5 + 2706834*x^4 - 3326843*x^3 + 25059307*x^2 - 15437955*x + 101631179, 1)
 

Normalized defining polynomial

\( x^{12} - 4 x^{11} + 115 x^{10} - 368 x^{9} + 5800 x^{8} - 14486 x^{7} + 163344 x^{6} - 302185 x^{5} + 2706834 x^{4} - 3326843 x^{3} + 25059307 x^{2} - 15437955 x + 101631179 \)

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:  \(33511670005535928548329=13^{10}\cdot 79^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $75.35$
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:  $13, 79$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1027=13\cdot 79\)
Dirichlet character group:    $\lbrace$$\chi_{1027}(1,·)$, $\chi_{1027}(1026,·)$, $\chi_{1027}(868,·)$, $\chi_{1027}(870,·)$, $\chi_{1027}(712,·)$, $\chi_{1027}(394,·)$, $\chi_{1027}(238,·)$, $\chi_{1027}(789,·)$, $\chi_{1027}(633,·)$, $\chi_{1027}(315,·)$, $\chi_{1027}(157,·)$, $\chi_{1027}(159,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{63773189341157721081299468489} a^{11} - \frac{29960798821409571664019971773}{63773189341157721081299468489} a^{10} - \frac{3310621783131444441283506914}{63773189341157721081299468489} a^{9} - \frac{19669091526899949414743125413}{63773189341157721081299468489} a^{8} + \frac{29119325626970569696121656548}{63773189341157721081299468489} a^{7} - \frac{21079308327777610005056741313}{63773189341157721081299468489} a^{6} + \frac{2416777653909596713481605197}{63773189341157721081299468489} a^{5} - \frac{31487458406939460946692662551}{63773189341157721081299468489} a^{4} + \frac{2296715736709756206455479316}{63773189341157721081299468489} a^{3} + \frac{8758181149576209329326862096}{63773189341157721081299468489} a^{2} + \frac{31858194603020902393402403133}{63773189341157721081299468489} a + \frac{3821164779149867015814408679}{63773189341157721081299468489}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{6}\times C_{210}$, which has order $10080$ (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:  \( 120.784031363 \) (assuming GRH)
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

\(\Q(\sqrt{-1027}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{-79}) \), 3.3.169.1, \(\Q(\sqrt{13}, \sqrt{-79})\), 6.0.183061929427.2, \(\Q(\zeta_{13})^+\), 6.0.14081686879.2

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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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
$13$13.6.5.2$x^{6} - 13$$6$$1$$5$$C_6$$[\ ]_{6}$
13.6.5.2$x^{6} - 13$$6$$1$$5$$C_6$$[\ ]_{6}$
$79$79.2.1.2$x^{2} + 158$$2$$1$$1$$C_2$$[\ ]_{2}$
79.2.1.2$x^{2} + 158$$2$$1$$1$$C_2$$[\ ]_{2}$
79.2.1.2$x^{2} + 158$$2$$1$$1$$C_2$$[\ ]_{2}$
79.2.1.2$x^{2} + 158$$2$$1$$1$$C_2$$[\ ]_{2}$
79.2.1.2$x^{2} + 158$$2$$1$$1$$C_2$$[\ ]_{2}$
79.2.1.2$x^{2} + 158$$2$$1$$1$$C_2$$[\ ]_{2}$