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Properties

Label 12.0.57001208768...5625.8
Degree $12$
Signature $[0, 6]$
Discriminant $3^{6}\cdot 5^{6}\cdot 7^{10}\cdot 11^{6}$
Root discriminant $65.01$
Ramified primes $3, 5, 7, 11$
Class number $5328$ (GRH)
Class group $[2, 2, 1332]$ (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![31120741, -2068070, 9700034, -402906, 1418605, -104688, 115149, -9320, 5088, -322, 113, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^11 + 113*x^10 - 322*x^9 + 5088*x^8 - 9320*x^7 + 115149*x^6 - 104688*x^5 + 1418605*x^4 - 402906*x^3 + 9700034*x^2 - 2068070*x + 31120741)
 
gp: K = bnfinit(x^12 - 4*x^11 + 113*x^10 - 322*x^9 + 5088*x^8 - 9320*x^7 + 115149*x^6 - 104688*x^5 + 1418605*x^4 - 402906*x^3 + 9700034*x^2 - 2068070*x + 31120741, 1)
 

Normalized defining polynomial

\( x^{12} - 4 x^{11} + 113 x^{10} - 322 x^{9} + 5088 x^{8} - 9320 x^{7} + 115149 x^{6} - 104688 x^{5} + 1418605 x^{4} - 402906 x^{3} + 9700034 x^{2} - 2068070 x + 31120741 \)

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:  \(5700120876856238765625=3^{6}\cdot 5^{6}\cdot 7^{10}\cdot 11^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $65.01$
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:  $3, 5, 7, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1155=3\cdot 5\cdot 7\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{1155}(1024,·)$, $\chi_{1155}(1,·)$, $\chi_{1155}(1154,·)$, $\chi_{1155}(131,·)$, $\chi_{1155}(164,·)$, $\chi_{1155}(331,·)$, $\chi_{1155}(461,·)$, $\chi_{1155}(529,·)$, $\chi_{1155}(626,·)$, $\chi_{1155}(694,·)$, $\chi_{1155}(824,·)$, $\chi_{1155}(991,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{1874} a^{8} + \frac{104}{937} a^{7} + \frac{253}{1874} a^{6} - \frac{151}{1874} a^{5} - \frac{198}{937} a^{4} + \frac{323}{1874} a^{3} - \frac{381}{1874} a^{2} + \frac{541}{1874} a + \frac{38}{937}$, $\frac{1}{1874} a^{9} + \frac{91}{1874} a^{7} - \frac{303}{1874} a^{6} - \frac{423}{937} a^{5} + \frac{235}{1874} a^{4} - \frac{101}{1874} a^{3} - \frac{793}{1874} a^{2} - \frac{6}{937} a - \frac{408}{937}$, $\frac{1}{1874} a^{10} + \frac{223}{937} a^{7} - \frac{222}{937} a^{6} + \frac{429}{937} a^{5} - \frac{304}{937} a^{4} + \frac{735}{1874} a^{3} + \frac{927}{1874} a^{2} - \frac{193}{937} a + \frac{290}{937}$, $\frac{1}{1069439853343973107845214} a^{11} + \frac{76563794108593237187}{534719926671986553922607} a^{10} + \frac{173377148859219457477}{1069439853343973107845214} a^{9} - \frac{122563206875767687848}{534719926671986553922607} a^{8} + \frac{151574311651046713078827}{1069439853343973107845214} a^{7} - \frac{77170979986293919848369}{534719926671986553922607} a^{6} + \frac{357604644259752231684933}{1069439853343973107845214} a^{5} - \frac{103352572478058506381839}{534719926671986553922607} a^{4} + \frac{143816350886648605851877}{1069439853343973107845214} a^{3} + \frac{165110655342118777128725}{534719926671986553922607} a^{2} - \frac{83383898047662115768374}{534719926671986553922607} a + \frac{440544984310065100858823}{1069439853343973107845214}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{1332}$, which has order $5328$ (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:  \( 104.88200347693757 \) (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

\(\Q(\sqrt{-231}) \), \(\Q(\sqrt{-1155}) \), \(\Q(\sqrt{5}) \), \(\Q(\zeta_{7})^+\), \(\Q(\sqrt{5}, \sqrt{-231})\), 6.0.603993159.1, 6.0.75499144875.3, 6.6.300125.1

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 R R R ${\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/29.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{4}$

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
$3$3.12.6.2$x^{12} + 108 x^{6} - 243 x^{2} + 2916$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$5$5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$7$7.12.10.1$x^{12} - 70 x^{6} + 35721$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$
$11$11.6.3.1$x^{6} - 22 x^{4} + 121 x^{2} - 11979$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
11.6.3.1$x^{6} - 22 x^{4} + 121 x^{2} - 11979$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$