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Properties

Label 12.0.7445055839159169.1
Degree $12$
Signature $[0, 6]$
Discriminant $3^{6}\cdot 7^{8}\cdot 11^{6}$
Root discriminant $21.02$
Ramified primes $3, 7, 11$
Class number $1$
Class group Trivial
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![5041, -284, -2043, 310, 545, -126, 31, 42, -16, 2, -3, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 - 3*x^10 + 2*x^9 - 16*x^8 + 42*x^7 + 31*x^6 - 126*x^5 + 545*x^4 + 310*x^3 - 2043*x^2 - 284*x + 5041)
 
gp: K = bnfinit(x^12 - x^11 - 3*x^10 + 2*x^9 - 16*x^8 + 42*x^7 + 31*x^6 - 126*x^5 + 545*x^4 + 310*x^3 - 2043*x^2 - 284*x + 5041, 1)
 

Normalized defining polynomial

\( x^{12} - x^{11} - 3 x^{10} + 2 x^{9} - 16 x^{8} + 42 x^{7} + 31 x^{6} - 126 x^{5} + 545 x^{4} + 310 x^{3} - 2043 x^{2} - 284 x + 5041 \)

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:  \(7445055839159169=3^{6}\cdot 7^{8}\cdot 11^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.02$
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:  $3, 7, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(231=3\cdot 7\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{231}(32,·)$, $\chi_{231}(1,·)$, $\chi_{231}(67,·)$, $\chi_{231}(100,·)$, $\chi_{231}(197,·)$, $\chi_{231}(65,·)$, $\chi_{231}(43,·)$, $\chi_{231}(109,·)$, $\chi_{231}(142,·)$, $\chi_{231}(23,·)$, $\chi_{231}(155,·)$, $\chi_{231}(221,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3613} a^{10} - \frac{317}{3613} a^{9} + \frac{313}{3613} a^{8} - \frac{496}{3613} a^{7} - \frac{1117}{3613} a^{6} + \frac{512}{3613} a^{5} - \frac{560}{3613} a^{4} + \frac{1088}{3613} a^{3} + \frac{931}{3613} a^{2} - \frac{1651}{3613} a - \frac{432}{3613}$, $\frac{1}{4595425147802341} a^{11} - \frac{89032330436}{4595425147802341} a^{10} + \frac{1423403121463475}{4595425147802341} a^{9} + \frac{1223338544135600}{4595425147802341} a^{8} - \frac{1181020182443515}{4595425147802341} a^{7} - \frac{651160092337744}{4595425147802341} a^{6} + \frac{537837269660764}{4595425147802341} a^{5} + \frac{294841058923763}{4595425147802341} a^{4} + \frac{1453578054492989}{4595425147802341} a^{3} + \frac{1072277202728848}{4595425147802341} a^{2} + \frac{338699711436188}{4595425147802341} a + \frac{19688489604339}{64724297856371}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( -\frac{171538516}{1271913962857} a^{11} - \frac{229320242}{1271913962857} a^{10} + \frac{236673978}{1271913962857} a^{9} + \frac{2659675842}{1271913962857} a^{8} + \frac{367897291}{1271913962857} a^{7} + \frac{5099088981}{1271913962857} a^{6} - \frac{12450229249}{1271913962857} a^{5} + \frac{1841497749}{1271913962857} a^{4} - \frac{977179529}{1271913962857} a^{3} - \frac{424329714659}{1271913962857} a^{2} + \frac{2261644935}{1271913962857} a + \frac{20200198995}{17914281167} \) (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:  \( 1236.8356247 \)
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

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{33}) \), \(\Q(\zeta_{7})^+\), \(\Q(\sqrt{-3}, \sqrt{-11})\), 6.0.64827.1, 6.0.3195731.1, 6.6.86284737.1

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 R ${\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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$3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$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}$
$11$11.12.6.1$x^{12} + 242 x^{8} + 21296 x^{6} + 14641 x^{4} + 1932612 x^{2} + 113379904$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$