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Properties

Label 12.12.1003808920...6704.1
Degree $12$
Signature $[12, 0]$
Discriminant $2^{24}\cdot 3^{10}\cdot 11^{20}\cdot 197^{4}$
Root discriminant $3163.28$
Ramified primes $2, 3, 11, 197$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $M_{12}$ (as 12T295)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3005497215363897, 167503033069380, -62489273434236, -2361270759804, 490521018900, 11122519452, -1841908992, -21476268, 3435883, 16940, -3036, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^11 - 3036*x^10 + 16940*x^9 + 3435883*x^8 - 21476268*x^7 - 1841908992*x^6 + 11122519452*x^5 + 490521018900*x^4 - 2361270759804*x^3 - 62489273434236*x^2 + 167503033069380*x + 3005497215363897)
 
gp: K = bnfinit(x^12 - 4*x^11 - 3036*x^10 + 16940*x^9 + 3435883*x^8 - 21476268*x^7 - 1841908992*x^6 + 11122519452*x^5 + 490521018900*x^4 - 2361270759804*x^3 - 62489273434236*x^2 + 167503033069380*x + 3005497215363897, 1)
 

Normalized defining polynomial

\( x^{12} - 4 x^{11} - 3036 x^{10} + 16940 x^{9} + 3435883 x^{8} - 21476268 x^{7} - 1841908992 x^{6} + 11122519452 x^{5} + 490521018900 x^{4} - 2361270759804 x^{3} - 62489273434236 x^{2} + 167503033069380 x + 3005497215363897 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $12$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1003808920860966325428537416433732056776704=2^{24}\cdot 3^{10}\cdot 11^{20}\cdot 197^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $3163.28$
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:  $2, 3, 11, 197$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4}$, $\frac{1}{3} a^{9} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{4}$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{7} + \frac{1}{3} a^{4}$, $\frac{1}{1381062463450314677309785141353293064274467677997552439431012291} a^{11} + \frac{83006382019768183191333538949396496468425209534786159213035059}{1381062463450314677309785141353293064274467677997552439431012291} a^{10} + \frac{62883459503669434361049758728590794028983457053154636630732266}{460354154483438225769928380451097688091489225999184146477004097} a^{9} - \frac{12511363826121536385441804300913603008513163198588757398555147}{1381062463450314677309785141353293064274467677997552439431012291} a^{8} + \frac{349996822644077599378577200870277232265020162260012169887430523}{1381062463450314677309785141353293064274467677997552439431012291} a^{7} - \frac{85390331002057024533612537873499574207746995399646971766097731}{460354154483438225769928380451097688091489225999184146477004097} a^{6} + \frac{216474848890327448307706605949343323756563872794588200715852888}{460354154483438225769928380451097688091489225999184146477004097} a^{5} - \frac{24995165600273939815968172092717379289948075082066950941712984}{460354154483438225769928380451097688091489225999184146477004097} a^{4} - \frac{166264901571010258977340992662247103591192515048522968660419312}{460354154483438225769928380451097688091489225999184146477004097} a^{3} - \frac{9985008638794515714298120552137983138024250415267806753375348}{153451384827812741923309460150365896030496408666394715492334699} a^{2} - \frac{64029408886394080567305230934526590646037925855121654268267116}{153451384827812741923309460150365896030496408666394715492334699} a + \frac{73541827634086676514550564036611713169470446284177156326679891}{153451384827812741923309460150365896030496408666394715492334699}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $11$
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:  \( 377943984207000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$M_{12}$ (as 12T295):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 95040
The 15 conjugacy class representatives for $M_{12}$
Character table for $M_{12}$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

Sibling fields

Degree 12 sibling: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R R ${\href{/LocalNumberField/5.10.0.1}{10} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$ ${\href{/LocalNumberField/7.11.0.1}{11} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ R ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.11.0.1}{11} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$2$2.12.24.146$x^{12} + 10 x^{10} + 6 x^{8} + 8 x^{7} + 16 x^{6} + 8 x^{4} + 16 x^{3} + 16 x^{2} + 16 x - 8$$4$$3$$24$12T60$[2, 2, 2, 3, 3]^{3}$
$3$3.4.3.2$x^{4} - 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
3.8.7.2$x^{8} - 3$$8$$1$$7$$QD_{16}$$[\ ]_{8}^{2}$
$11$$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
11.11.20.8$x^{11} - 11 x^{10} + 1221$$11$$1$$20$$C_{11}$$[2]$
$197$197.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
197.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
197.4.2.1$x^{4} + 985 x^{2} + 349281$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
197.4.2.1$x^{4} + 985 x^{2} + 349281$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$