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Properties

Label 12.12.1242206740...0000.1
Degree $12$
Signature $[12, 0]$
Discriminant $2^{8}\cdot 3^{20}\cdot 5^{10}\cdot 31^{6}\cdot 43^{6}\cdot 347^{2}\cdot 145242751^{2}$
Root discriminant $84{,}045.83$
Ramified primes $2, 3, 5, 31, 43, 347, 145242751$
Class number $8$ (GRH)
Class group $[2, 4]$ (GRH)
Galois group $A_{12}$ (as 12T300)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![11016133065262452400, -10416503841795589200, -5991420705430217400, -9062974515513700, 7206776378221275, 7515402970950, -3123608892765, -466244190, 579490200, -150220, -43095, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 43095*x^10 - 150220*x^9 + 579490200*x^8 - 466244190*x^7 - 3123608892765*x^6 + 7515402970950*x^5 + 7206776378221275*x^4 - 9062974515513700*x^3 - 5991420705430217400*x^2 - 10416503841795589200*x + 11016133065262452400)
 
gp: K = bnfinit(x^12 - 43095*x^10 - 150220*x^9 + 579490200*x^8 - 466244190*x^7 - 3123608892765*x^6 + 7515402970950*x^5 + 7206776378221275*x^4 - 9062974515513700*x^3 - 5991420705430217400*x^2 - 10416503841795589200*x + 11016133065262452400, 1)
 

Normalized defining polynomial

\( x^{12} - 43095 x^{10} - 150220 x^{9} + 579490200 x^{8} - 466244190 x^{7} - 3123608892765 x^{6} + 7515402970950 x^{5} + 7206776378221275 x^{4} - 9062974515513700 x^{3} - 5991420705430217400 x^{2} - 10416503841795589200 x + 11016133065262452400 \)

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:  \(124220674037581049430090394518362216959104955545802500000000=2^{8}\cdot 3^{20}\cdot 5^{10}\cdot 31^{6}\cdot 43^{6}\cdot 347^{2}\cdot 145242751^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $84{,}045.83$
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:  $2, 3, 5, 31, 43, 347, 145242751$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{10} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{10} a^{7} - \frac{1}{2} a$, $\frac{1}{20} a^{8} - \frac{1}{2} a^{5} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{120} a^{9} - \frac{1}{40} a^{8} - \frac{1}{20} a^{7} - \frac{1}{20} a^{6} + \frac{1}{4} a^{5} - \frac{3}{8} a^{3} + \frac{3}{8} a^{2} - \frac{1}{2} a + \frac{1}{6}$, $\frac{1}{360} a^{10} - \frac{1}{360} a^{9} + \frac{1}{60} a^{7} - \frac{1}{60} a^{6} - \frac{1}{2} a^{5} - \frac{1}{8} a^{4} - \frac{1}{8} a^{3} + \frac{1}{4} a^{2} - \frac{5}{18} a - \frac{2}{9}$, $\frac{1}{37859038640632892408341363959161713884377076143828727713340238520296224602743136523600} a^{11} + \frac{95194257971737122561616669123685700528594371100422470988898030718247002142765911}{291223374158714556987241261224320876033669816490990213179540296309970958482639511720} a^{10} + \frac{33429469139619723240798490292574141984854273791407145525838123189362426158148289}{582446748317429113974482522448641752067339632981980426359080592619941916965279023440} a^{9} + \frac{9253295586397044527126216613263589834880764444787807525165823435975023760409683549}{1261967954687763080278045465305390462812569204794290923778007950676540820091437884120} a^{8} + \frac{20972982533466362088550182620526813289590365281530837903348841639660221292763670691}{630983977343881540139022732652695231406284602397145461889003975338270410045718942060} a^{7} - \frac{4684705575427280324239704469879236683843376263818167898915294121545962390782299685}{252393590937552616055609093061078092562513840958858184755601590135308164018287576824} a^{6} + \frac{50068798801043336013927411301074501181502024230392933329869228844181303550203133379}{841311969791842053518696976870260308541712803196193949185338633784360546727625256080} a^{5} + \frac{130266295339391332716236003483645287937341078031638706785650167389662847612761283}{489134866158047705534126149343174597989367908834996482084499205688581713213735614} a^{4} - \frac{5115408330696354689221437004007550625592281067360074525727541076965570843345648217}{168262393958368410703739395374052061708342560639238789837067726756872109345525051216} a^{3} - \frac{253921163639682909649608235955490275739731105796142655917254755822229798821465053511}{757180772812657848166827279183234277687541522876574554266804770405924492054862730472} a^{2} - \frac{111786254341035508096136561829592985680760796238430931509908030539067453874118091027}{378590386406328924083413639591617138843770761438287277133402385202962246027431365236} a + \frac{5941988054537275839923297288276497381317258863019381969270818718053292579533115055}{14561168707935727849362063061216043801683490824549510658977014815498547924131975586}$

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

Class group and class number

$C_{2}\times C_{4}$, which has order $8$ (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:  \( 105166289261000000000000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$A_{12}$ (as 12T300):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 239500800
The 43 conjugacy class representatives for $A_{12}$
Character table for $A_{12}$ is not computed

Intermediate fields

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

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 R ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.9.0.1}{9} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.9.0.1}{9} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ R ${\href{/LocalNumberField/37.5.0.1}{5} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.11.0.1}{11} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ R ${\href{/LocalNumberField/47.11.0.1}{11} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.11.0.1}{11} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
$3$$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.9.20.54$x^{9} + 12 x^{6} + 24 x^{3} + 3$$9$$1$$20$$((C_3^3:C_3):C_2):C_2$$[3/2, 5/2, 8/3, 17/6]_{2}^{2}$
$5$5.12.10.1$x^{12} + 6 x^{11} + 27 x^{10} + 80 x^{9} + 195 x^{8} + 366 x^{7} + 571 x^{6} + 702 x^{5} + 1005 x^{4} + 1140 x^{3} + 357 x^{2} - 138 x + 44$$6$$2$$10$$D_6$$[\ ]_{6}^{2}$
$31$31.4.2.2$x^{4} - 31 x^{2} + 11532$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
31.8.4.2$x^{8} - 59582 x^{2} + 15699857$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
43Data not computed
347Data not computed
145242751Data not computed