↑

Properties

Label 12.12.8039530502...0000.1
Degree $12$
Signature $[12, 0]$
Discriminant $2^{8}\cdot 3^{16}\cdot 5^{14}\cdot 7^{8}\cdot 137^{6}\cdot 5599914953^{2}$
Root discriminant $81{,}053.02$
Ramified primes $2, 3, 5, 7, 137, 5599914953$
Class number $8$ (GRH)
Class group $[2, 4]$ (GRH)
Galois group $A_{12}$ (as 12T300)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![64178007246927738000, 43526001864084588000, 9912416687818565400, 906390235169823000, 24196102698409425, -551549392646400, -11816325894450, 55149567840, 1413391275, -1521160, -64290, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 64290*x^10 - 1521160*x^9 + 1413391275*x^8 + 55149567840*x^7 - 11816325894450*x^6 - 551549392646400*x^5 + 24196102698409425*x^4 + 906390235169823000*x^3 + 9912416687818565400*x^2 + 43526001864084588000*x + 64178007246927738000)
 
gp: K = bnfinit(x^12 - 64290*x^10 - 1521160*x^9 + 1413391275*x^8 + 55149567840*x^7 - 11816325894450*x^6 - 551549392646400*x^5 + 24196102698409425*x^4 + 906390235169823000*x^3 + 9912416687818565400*x^2 + 43526001864084588000*x + 64178007246927738000, 1)
 

Normalized defining polynomial

\( x^{12} - 64290 x^{10} - 1521160 x^{9} + 1413391275 x^{8} + 55149567840 x^{7} - 11816325894450 x^{6} - 551549392646400 x^{5} + 24196102698409425 x^{4} + 906390235169823000 x^{3} + 9912416687818565400 x^{2} + 43526001864084588000 x + 64178007246927738000 \)

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:  \(80395305020560195610021041260875524780564497501562500000000=2^{8}\cdot 3^{16}\cdot 5^{14}\cdot 7^{8}\cdot 137^{6}\cdot 5599914953^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $81{,}053.02$
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:  $2, 3, 5, 7, 137, 5599914953$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{40} a^{5} - \frac{1}{4} a^{4} - \frac{1}{8} a^{3} - \frac{1}{4} a^{2} - \frac{3}{8} a + \frac{1}{4}$, $\frac{1}{560} a^{6} - \frac{45}{112} a^{4} + \frac{9}{28} a^{3} - \frac{23}{112} a^{2} - \frac{3}{28} a + \frac{5}{28}$, $\frac{1}{8400} a^{7} + \frac{1}{1680} a^{6} - \frac{1}{560} a^{5} + \frac{19}{80} a^{4} - \frac{11}{112} a^{3} - \frac{257}{560} a^{2} + \frac{15}{56} a - \frac{5}{14}$, $\frac{1}{16800} a^{8} - \frac{1}{16800} a^{7} - \frac{1}{1120} a^{5} + \frac{27}{70} a^{4} + \frac{193}{1120} a^{3} - \frac{53}{1120} a^{2} + \frac{13}{56} a + \frac{5}{56}$, $\frac{1}{16800} a^{9} - \frac{1}{16800} a^{7} - \frac{1}{1120} a^{6} + \frac{11}{1120} a^{5} + \frac{69}{224} a^{4} - \frac{73}{1120} a^{2} - \frac{3}{56} a + \frac{19}{56}$, $\frac{1}{100800} a^{10} - \frac{1}{50400} a^{9} - \frac{1}{50400} a^{8} - \frac{1}{16800} a^{7} + \frac{1}{2240} a^{6} - \frac{3}{1120} a^{5} + \frac{17}{840} a^{4} - \frac{269}{1680} a^{3} + \frac{3337}{6720} a^{2} + \frac{3}{112} a + \frac{39}{112}$, $\frac{1}{9765557535478334155453447748579665103813592217255743869842760616307200} a^{11} - \frac{11199729762230301023293837693482995145239441467689403659465210793}{4882778767739167077726723874289832551906796108627871934921380308153600} a^{10} - \frac{119324899843147109203132263529243684595435867730719648851270339519}{4882778767739167077726723874289832551906796108627871934921380308153600} a^{9} - \frac{81370178681022285685142914668983667396896142187976129162103001}{2980939418644180145132310057564000336939435963753279569549072227200} a^{8} + \frac{62401590595114580941055357718878756221425938202668114785797903947}{1085061948386481572828160860953296122645954690806193763315862290700800} a^{7} + \frac{5550450462717178352854467290007097932990764717180029001972437711}{15500884976949736754688012299332801752085067011517053761655175581440} a^{6} + \frac{416692494458548277452886831468655141566376492847842317152678587701}{46502654930849210264064036897998405256255201034551161284965526744320} a^{5} - \frac{3978782282637128136530690110261680825050676473038699176609747055903}{32551858451594447184844825828598883679378640724185812899475868721024} a^{4} - \frac{259111550171248026639914450837003395102828124665422076237140033377033}{651037169031888943696896516571977673587572814483716257989517374420480} a^{3} - \frac{1638895811320368695780164436317896120661327282407706707167308426369}{12056243870960906364757342899481068029399496564513264036842914341120} a^{2} - \frac{52567928603764423888253791153612958568754809019704755089936958441}{516696165898324558489600409977760058402835567050568458721839186048} a + \frac{594153128810207086417907816919330816717060432435417917027334457355}{1808436580644135954713601434922160204409924484676989605526437151168}$

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:  \( 141592770050000000000000000 \) (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 R ${\href{/LocalNumberField/11.7.0.1}{7} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.11.0.1}{11} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.11.0.1}{11} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }$ ${\href{/LocalNumberField/29.9.0.1}{9} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }$ ${\href{/LocalNumberField/31.7.0.1}{7} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.9.0.1}{9} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.11.0.1}{11} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.11.0.1}{11} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$$\Q_{2}$$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}$
3.3.5.2$x^{3} + 21$$3$$1$$5$$S_3$$[5/2]_{2}$
3.6.11.5$x^{6} + 3 x^{3} + 6$$6$$1$$11$$S_3^2$$[2, 5/2]_{2}^{2}$
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.10.13.4$x^{10} + 5 x^{4} + 5$$10$$1$$13$$F_{5}\times C_2$$[3/2]_{2}^{4}$
$7$$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
7.5.4.1$x^{5} - 7$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
7.5.4.1$x^{5} - 7$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
$137$137.12.6.1$x^{12} + 149138474 x^{6} - 48261724457 x^{2} + 5560571106762169$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
5599914953Data not computed