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Properties

Label 12.0.101439097511492601.1
Degree $12$
Signature $[0, 6]$
Discriminant $3^{6}\cdot 7^{8}\cdot 17^{6}$
Root discriminant $26.13$
Ramified primes $3, 7, 17$
Class number $13$
Class group $[13]$
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![169, 325, 1366, -1139, 3303, -210, 1039, -98, 243, -5, 18, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 + 18*x^10 - 5*x^9 + 243*x^8 - 98*x^7 + 1039*x^6 - 210*x^5 + 3303*x^4 - 1139*x^3 + 1366*x^2 + 325*x + 169)
 
gp: K = bnfinit(x^12 - x^11 + 18*x^10 - 5*x^9 + 243*x^8 - 98*x^7 + 1039*x^6 - 210*x^5 + 3303*x^4 - 1139*x^3 + 1366*x^2 + 325*x + 169, 1)
 

Normalized defining polynomial

\( x^{12} - x^{11} + 18 x^{10} - 5 x^{9} + 243 x^{8} - 98 x^{7} + 1039 x^{6} - 210 x^{5} + 3303 x^{4} - 1139 x^{3} + 1366 x^{2} + 325 x + 169 \)

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:  \(101439097511492601=3^{6}\cdot 7^{8}\cdot 17^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.13$
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:  $3, 7, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(357=3\cdot 7\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{357}(256,·)$, $\chi_{357}(1,·)$, $\chi_{357}(67,·)$, $\chi_{357}(137,·)$, $\chi_{357}(205,·)$, $\chi_{357}(239,·)$, $\chi_{357}(16,·)$, $\chi_{357}(305,·)$, $\chi_{357}(50,·)$, $\chi_{357}(86,·)$, $\chi_{357}(169,·)$, $\chi_{357}(254,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{13} a^{7} - \frac{4}{13} a^{4} - \frac{5}{13} a$, $\frac{1}{13} a^{8} - \frac{4}{13} a^{5} - \frac{5}{13} a^{2}$, $\frac{1}{13} a^{9} - \frac{4}{13} a^{6} - \frac{5}{13} a^{3}$, $\frac{1}{1651} a^{10} - \frac{34}{1651} a^{9} + \frac{51}{1651} a^{8} + \frac{32}{1651} a^{7} - \frac{345}{1651} a^{6} + \frac{823}{1651} a^{5} + \frac{462}{1651} a^{4} + \frac{157}{1651} a^{3} + \frac{83}{1651} a^{2} + \frac{28}{1651} a - \frac{12}{127}$, $\frac{1}{142981286048159} a^{11} - \frac{19225152281}{142981286048159} a^{10} - \frac{1677804807688}{142981286048159} a^{9} - \frac{1563497340342}{142981286048159} a^{8} - \frac{536612629383}{142981286048159} a^{7} - \frac{60883127100071}{142981286048159} a^{6} - \frac{387337876531}{10998560465243} a^{5} - \frac{40367944636112}{142981286048159} a^{4} - \frac{23457296565359}{142981286048159} a^{3} - \frac{11535201753722}{142981286048159} a^{2} - \frac{1105937483473}{10998560465243} a - \frac{332525799508}{846043112711}$

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

Class group and class number

$C_{13}$, which has order $13$

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:  \( -\frac{1786904954}{1125836898017} a^{11} + \frac{2591583648}{1125836898017} a^{10} - \frac{32521351238}{1125836898017} a^{9} + \frac{22762679686}{1125836898017} a^{8} - \frac{430368660023}{1125836898017} a^{7} + \frac{366338661178}{1125836898017} a^{6} - \frac{140978502354}{86602838309} a^{5} + \frac{1136094959356}{1125836898017} a^{4} - \frac{5667060866394}{1125836898017} a^{3} + \frac{4730235424204}{1125836898017} a^{2} - \frac{164909029884}{86602838309} a + \frac{3778101786}{6661756793} \) (order $6$)
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:  \( 1059.54542703 \)
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

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-51}) \), \(\Q(\zeta_{7})^+\), \(\Q(\sqrt{-3}, \sqrt{17})\), 6.0.64827.1, 6.6.11796113.1, 6.0.318495051.1

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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/13.1.0.1}{1} }^{12}$ R ${\href{/LocalNumberField/19.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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$$[\ ]_{2}^{6}$
$7$7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
$17$17.12.6.1$x^{12} + 117912 x^{6} - 1419857 x^{2} + 3475809936$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$