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Properties

Label 12.12.3929172074...0000.1
Degree $12$
Signature $[12, 0]$
Discriminant $2^{16}\cdot 3^{4}\cdot 5^{8}\cdot 11^{12}\cdot 13^{4}\cdot 17^{4}\cdot 347^{10}$
Root discriminant $92{,}510.65$
Ramified primes $2, 3, 5, 11, 13, 17, 347$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $M_{11}$ (as 12T272)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-861259903428927255084, 31953284631467426916, 69523530327654696348, -3552570264974645508, 14413851909825665, 1461301745883639, -16041513207467, -152359965054, 2259444770, 4532198, -100067, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 - 100067*x^10 + 4532198*x^9 + 2259444770*x^8 - 152359965054*x^7 - 16041513207467*x^6 + 1461301745883639*x^5 + 14413851909825665*x^4 - 3552570264974645508*x^3 + 69523530327654696348*x^2 + 31953284631467426916*x - 861259903428927255084)
 
gp: K = bnfinit(x^12 - x^11 - 100067*x^10 + 4532198*x^9 + 2259444770*x^8 - 152359965054*x^7 - 16041513207467*x^6 + 1461301745883639*x^5 + 14413851909825665*x^4 - 3552570264974645508*x^3 + 69523530327654696348*x^2 + 31953284631467426916*x - 861259903428927255084, 1)
 

Normalized defining polynomial

\( x^{12} - x^{11} - 100067 x^{10} + 4532198 x^{9} + 2259444770 x^{8} - 152359965054 x^{7} - 16041513207467 x^{6} + 1461301745883639 x^{5} + 14413851909825665 x^{4} - 3552570264974645508 x^{3} + 69523530327654696348 x^{2} + 31953284631467426916 x - 861259903428927255084 \)

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:  \(392917207432060604272953142193224798570962446217446400000000=2^{16}\cdot 3^{4}\cdot 5^{8}\cdot 11^{12}\cdot 13^{4}\cdot 17^{4}\cdot 347^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $92{,}510.65$
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:  $2, 3, 5, 11, 13, 17, 347$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{10} a^{6} + \frac{1}{5} a^{5} + \frac{1}{5} a^{4} - \frac{1}{2} a^{3} - \frac{1}{5} a^{2} + \frac{1}{5} a + \frac{2}{5}$, $\frac{1}{110} a^{7} - \frac{3}{110} a^{6} + \frac{1}{5} a^{5} - \frac{7}{22} a^{4} - \frac{17}{110} a^{3} - \frac{9}{55} a^{2} + \frac{27}{55} a + \frac{4}{11}$, $\frac{1}{660} a^{8} + \frac{2}{55} a^{6} - \frac{28}{165} a^{5} + \frac{1}{66} a^{4} + \frac{103}{330} a^{3} + \frac{13}{60} a^{2} + \frac{19}{110} a + \frac{9}{110}$, $\frac{1}{660} a^{9} + \frac{13}{330} a^{6} - \frac{14}{165} a^{5} - \frac{71}{330} a^{4} + \frac{221}{660} a^{3} + \frac{7}{55} a^{2} + \frac{7}{22} a - \frac{3}{55}$, $\frac{1}{660} a^{10} + \frac{1}{330} a^{7} + \frac{4}{165} a^{6} - \frac{1}{66} a^{5} - \frac{259}{660} a^{4} - \frac{14}{55} a^{3} - \frac{3}{110} a^{2} - \frac{1}{55} a - \frac{5}{11}$, $\frac{1}{254042285050758793186851117572621044761116431346977115215377265953412716315343623772699309140} a^{11} + \frac{22025401578906965778176086352021159483591875644077136795521637822514852101824139278516993}{50808457010151758637370223514524208952223286269395423043075453190682543263068724754539861828} a^{10} - \frac{13195612641300628080655191267813057357923055535695629111218479670976893834948148853271906}{63510571262689698296712779393155261190279107836744278803844316488353179078835905943174827285} a^{9} - \frac{14519409426429798788680906712477403609593757302437194787819491448527313218319924845637949}{25404228505075879318685111757262104476111643134697711521537726595341271631534362377269930914} a^{8} + \frac{172828622653005147619405365034939976695958698135438480977251964283496502073525158384326279}{42340380841793132197808519595436840793519405224496185869229544325568786052557270628783218190} a^{7} + \frac{287754759489797263504637704679386726524489811904043709245535246213126206666312866441380111}{7471831913257611564319150516841795434150483274911091623981684292747432832804224228608803210} a^{6} - \frac{6703712666913377607582436614804341143256639477744921365558650876045234131071316490830662699}{50808457010151758637370223514524208952223286269395423043075453190682543263068724754539861828} a^{5} + \frac{22551364318849558157578619599174130044035532345173573669391272740159263115593013617363014769}{50808457010151758637370223514524208952223286269395423043075453190682543263068724754539861828} a^{4} + \frac{2313422180040137242944755473968400008380720619008177098098486151735306119521634845814281333}{25404228505075879318685111757262104476111643134697711521537726595341271631534362377269930914} a^{3} + \frac{4379319149032117044520897051174453257611868489379094046441159251150561970609438871590593173}{63510571262689698296712779393155261190279107836744278803844316488353179078835905943174827285} a^{2} + \frac{114245483135950200191827056304032306239040456554123777573555702868095877645130379987466726}{1245305318876268594053191752806965905691747212485181937330280715457905472134037371434800535} a - \frac{528003584162950389530637843833433919062165898768855499660873583991234456976833558676288331}{1245305318876268594053191752806965905691747212485181937330280715457905472134037371434800535}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 976514024364000000000000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$M_{11}$ (as 12T272):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7920
The 10 conjugacy class representatives for $M_{11}$
Character table for $M_{11}$

Intermediate fields

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

Sibling fields

Degree 11 sibling: data not computed
Degree 22 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 R ${\href{/LocalNumberField/7.11.0.1}{11} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ R R R ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.11.0.1}{11} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.11.0.1}{11} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/47.11.0.1}{11} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.4.8.5$x^{4} + 2 x^{2} + 4 x + 6$$4$$1$$8$$D_{4}$$[2, 3]^{2}$
2.4.6.6$x^{4} - 20$$2$$2$$6$$D_{4}$$[2, 3]^{2}$
$3$3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
$5$5.4.2.2$x^{4} - 5 x^{2} + 50$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
5.8.6.3$x^{8} + 25 x^{4} + 200$$4$$2$$6$$C_8$$[\ ]_{4}^{2}$
$11$$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
11.11.12.4$x^{11} + 110 x^{2} + 11$$11$$1$$12$$C_{11}:C_5$$[6/5]_{5}$
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.3.0.1$x^{3} - 2 x + 6$$1$$3$$0$$C_3$$[\ ]^{3}$
13.6.3.1$x^{6} - 52 x^{4} + 676 x^{2} - 79092$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$17$$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
347Data not computed