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Properties

Label 11.3.38739734030...0000.1
Degree $11$
Signature $[3, 4]$
Discriminant $2^{18}\cdot 3^{8}\cdot 5^{6}\cdot 7^{8}\cdot 251^{6}$
Root discriminant $1394.37$
Ramified primes $2, 3, 5, 7, 251$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $M_{11}$ (as 11T6)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![95047524720, -46920135888, 10531148484, -762562533, 136254876, -5337282, 643152, -15012, -156, -318, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^11 - 4*x^10 - 318*x^9 - 156*x^8 - 15012*x^7 + 643152*x^6 - 5337282*x^5 + 136254876*x^4 - 762562533*x^3 + 10531148484*x^2 - 46920135888*x + 95047524720)
 
gp: K = bnfinit(x^11 - 4*x^10 - 318*x^9 - 156*x^8 - 15012*x^7 + 643152*x^6 - 5337282*x^5 + 136254876*x^4 - 762562533*x^3 + 10531148484*x^2 - 46920135888*x + 95047524720, 1)
 

Normalized defining polynomial

\( x^{11} - 4 x^{10} - 318 x^{9} - 156 x^{8} - 15012 x^{7} + 643152 x^{6} - 5337282 x^{5} + 136254876 x^{4} - 762562533 x^{3} + 10531148484 x^{2} - 46920135888 x + 95047524720 \)

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

Invariants

Degree:  $11$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[3, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(38739734030642940596585926656000000=2^{18}\cdot 3^{8}\cdot 5^{6}\cdot 7^{8}\cdot 251^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1394.37$
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:  $2, 3, 5, 7, 251$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{10} a^{6} + \frac{1}{10} a^{5} + \frac{1}{5} a^{4} + \frac{2}{5} a^{3} + \frac{3}{10} a^{2} - \frac{1}{10} a$, $\frac{1}{10} a^{7} + \frac{1}{10} a^{5} + \frac{1}{5} a^{4} - \frac{1}{10} a^{3} - \frac{2}{5} a^{2} + \frac{1}{10} a$, $\frac{1}{100} a^{8} + \frac{1}{50} a^{7} - \frac{3}{100} a^{6} - \frac{1}{10} a^{5} + \frac{1}{20} a^{4} + \frac{2}{25} a^{3} + \frac{1}{100} a^{2} - \frac{1}{25} a - \frac{2}{5}$, $\frac{1}{300} a^{9} - \frac{1}{300} a^{8} - \frac{3}{100} a^{7} + \frac{3}{100} a^{6} + \frac{3}{20} a^{5} + \frac{21}{100} a^{4} - \frac{11}{100} a^{3} - \frac{9}{100} a^{2} + \frac{1}{25} a + \frac{2}{5}$, $\frac{1}{26089391730579751461608039800131575403300} a^{10} - \frac{1995198525489703034364758688184183487}{8696463910193250487202679933377191801100} a^{9} + \frac{679989249289484470385127053629334654}{176279673855268590956811079730618752725} a^{8} + \frac{297264654503737141448205818238721733997}{8696463910193250487202679933377191801100} a^{7} - \frac{51372607411409645491046008272951791164}{2174115977548312621800669983344297950275} a^{6} + \frac{1817140833035922942826531575510750208971}{8696463910193250487202679933377191801100} a^{5} + \frac{593244280491694857879639882045546121477}{4348231955096625243601339966688595900550} a^{4} + \frac{3936629591287183942971881409973624257657}{8696463910193250487202679933377191801100} a^{3} + \frac{3648907462373621760238858010842014151111}{8696463910193250487202679933377191801100} a^{2} + \frac{2014193167795343111425542388281273720091}{4348231955096625243601339966688595900550} a - \frac{136424346641283131076356526013811612974}{434823195509662524360133996668859590055}$

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:  $6$
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:  \( 513874226540000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$M_{11}$ (as 11T6):

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 12 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 R ${\href{/LocalNumberField/11.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/29.11.0.1}{11} }$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.11.0.1}{11} }$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.8.16.74$x^{8} + 14 x^{6} + 20$$8$$1$$16$$\textrm{GL(2,3)}$$[4/3, 4/3, 3]_{3}^{2}$
$3$$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.8.7.2$x^{8} - 3$$8$$1$$7$$QD_{16}$$[\ ]_{8}^{2}$
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
$7$$\Q_{7}$$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}$
251Data not computed