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Properties

Label 12.0.88620023623...0873.1
Degree $12$
Signature $[0, 6]$
Discriminant $7^{10}\cdot 73^{11}$
Root discriminant $258.40$
Ramified primes $7, 73$
Class number $1147854$ (GRH)
Class group $[1147854]$ (GRH)
Galois group $C_{12}$ (as 12T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![276287488, 235215360, 84181376, 6762912, 4161656, 605452, 216950, 5787, 3865, 362, 40, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 + 40*x^10 + 362*x^9 + 3865*x^8 + 5787*x^7 + 216950*x^6 + 605452*x^5 + 4161656*x^4 + 6762912*x^3 + 84181376*x^2 + 235215360*x + 276287488)
 
gp: K = bnfinit(x^12 - x^11 + 40*x^10 + 362*x^9 + 3865*x^8 + 5787*x^7 + 216950*x^6 + 605452*x^5 + 4161656*x^4 + 6762912*x^3 + 84181376*x^2 + 235215360*x + 276287488, 1)
 

Normalized defining polynomial

\( x^{12} - x^{11} + 40 x^{10} + 362 x^{9} + 3865 x^{8} + 5787 x^{7} + 216950 x^{6} + 605452 x^{5} + 4161656 x^{4} + 6762912 x^{3} + 84181376 x^{2} + 235215360 x + 276287488 \)

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:  \(88620023623867115145360460873=7^{10}\cdot 73^{11}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $258.40$
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:  $7, 73$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(511=7\cdot 73\)
Dirichlet character group:    $\lbrace$$\chi_{511}(1,·)$, $\chi_{511}(3,·)$, $\chi_{511}(9,·)$, $\chi_{511}(429,·)$, $\chi_{511}(143,·)$, $\chi_{511}(81,·)$, $\chi_{511}(243,·)$, $\chi_{511}(341,·)$, $\chi_{511}(265,·)$, $\chi_{511}(218,·)$, $\chi_{511}(27,·)$, $\chi_{511}(284,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{4} + \frac{1}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{16} a^{5} + \frac{1}{16} a^{3} + \frac{1}{8} a^{2}$, $\frac{1}{32} a^{6} + \frac{1}{32} a^{4} + \frac{1}{16} a^{3}$, $\frac{1}{128} a^{7} - \frac{1}{128} a^{6} + \frac{3}{128} a^{5} - \frac{7}{128} a^{4} - \frac{1}{32} a^{2} - \frac{1}{8} a$, $\frac{1}{9728} a^{8} - \frac{29}{9728} a^{7} - \frac{145}{9728} a^{6} - \frac{219}{9728} a^{5} - \frac{75}{2432} a^{4} + \frac{279}{2432} a^{3} - \frac{49}{304} a^{2} + \frac{47}{152} a + \frac{2}{19}$, $\frac{1}{77824} a^{9} + \frac{115}{38912} a^{7} - \frac{21}{9728} a^{6} - \frac{1179}{77824} a^{5} + \frac{5}{1216} a^{4} + \frac{1163}{19456} a^{3} - \frac{415}{2432} a^{2} - \frac{255}{1216} a - \frac{37}{152}$, $\frac{1}{466944} a^{10} - \frac{1}{233472} a^{9} + \frac{7}{233472} a^{8} - \frac{415}{116736} a^{7} - \frac{2001}{155648} a^{6} - \frac{2977}{233472} a^{5} - \frac{2861}{116736} a^{4} + \frac{2917}{58368} a^{3} + \frac{185}{1824} a^{2} + \frac{1567}{3648} a + \frac{49}{456}$, $\frac{1}{3550848544995320070144} a^{11} + \frac{208531333500043}{591808090832553345024} a^{10} - \frac{82881517135093}{18303343015439794176} a^{9} - \frac{30344630724595}{5191299042390818816} a^{8} + \frac{7307910975390530557}{3550848544995320070144} a^{7} + \frac{7966347707110215065}{1775424272497660035072} a^{6} + \frac{3769989393058163335}{295904045416276672512} a^{5} + \frac{4574870865056093483}{443856068124415008768} a^{4} - \frac{560247652073287367}{36988005677034584064} a^{3} + \frac{97374229506662489}{9247001419258646016} a^{2} + \frac{2661599110101115531}{6935251064443984512} a - \frac{328911771704951521}{866906383055498064}$

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

Class group and class number

$C_{1147854}$, which has order $1147854$ (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:  \( 10182106.529139174 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{12}$ (as 12T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 12
The 12 conjugacy class representatives for $C_{12}$
Character table for $C_{12}$

Intermediate fields

\(\Q(\sqrt{73}) \), 3.3.261121.1, 4.0.19061833.1, 6.6.4977444894793.1

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 ${\href{/LocalNumberField/2.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/3.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/5.12.0.1}{12} }$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/13.12.0.1}{12} }$ ${\href{/LocalNumberField/17.12.0.1}{12} }$ ${\href{/LocalNumberField/19.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/29.12.0.1}{12} }$ ${\href{/LocalNumberField/31.12.0.1}{12} }$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/47.12.0.1}{12} }$ ${\href{/LocalNumberField/53.12.0.1}{12} }$ ${\href{/LocalNumberField/59.12.0.1}{12} }$

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
$7$7.12.10.5$x^{12} + 56 x^{6} + 1323$$6$$2$$10$$C_{12}$$[\ ]_{6}^{2}$
$73$73.12.11.2$x^{12} - 1825$$12$$1$$11$$C_{12}$$[\ ]_{12}$