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Properties

Label 11.3.12102729797...9616.3
Degree $11$
Signature $[3, 4]$
Discriminant $2^{12}\cdot 3^{12}\cdot 11^{18}$
Root discriminant $357.26$
Ramified primes $2, 3, 11$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $A_{11}$ (as 11T7)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![152567584, -18249792, -1054240, 3734544, -1603536, -375936, 81048, 16104, -726, -220, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^11 - 220*x^9 - 726*x^8 + 16104*x^7 + 81048*x^6 - 375936*x^5 - 1603536*x^4 + 3734544*x^3 - 1054240*x^2 - 18249792*x + 152567584)
 
gp: K = bnfinit(x^11 - 220*x^9 - 726*x^8 + 16104*x^7 + 81048*x^6 - 375936*x^5 - 1603536*x^4 + 3734544*x^3 - 1054240*x^2 - 18249792*x + 152567584, 1)
 

Normalized defining polynomial

\( x^{11} - 220 x^{9} - 726 x^{8} + 16104 x^{7} + 81048 x^{6} - 375936 x^{5} - 1603536 x^{4} + 3734544 x^{3} - 1054240 x^{2} - 18249792 x + 152567584 \)

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:  \(12102729797630463961063919616=2^{12}\cdot 3^{12}\cdot 11^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $357.26$
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:  $2, 3, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{2} a^{4}$, $\frac{1}{4} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{8} a^{6} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{48} a^{7} - \frac{1}{24} a^{6} - \frac{1}{24} a^{4} - \frac{1}{6} a^{3} - \frac{1}{6} a - \frac{1}{6}$, $\frac{1}{48} a^{8} + \frac{1}{24} a^{6} - \frac{1}{24} a^{5} - \frac{1}{12} a^{3} - \frac{1}{6} a^{2} - \frac{1}{3}$, $\frac{1}{192} a^{9} - \frac{1}{96} a^{8} - \frac{1}{96} a^{7} + \frac{1}{96} a^{6} - \frac{1}{24} a^{5} - \frac{11}{48} a^{4} + \frac{1}{24} a^{3} + \frac{11}{24} a^{2} + \frac{1}{12} a + \frac{1}{12}$, $\frac{1}{101690150168102807551690368} a^{10} - \frac{52485768355834071406121}{25422537542025701887922592} a^{9} + \frac{90945935440015336077079}{16948358361350467925281728} a^{8} + \frac{109559543022718046728309}{16948358361350467925281728} a^{7} + \frac{320952950439188477185759}{8474179180675233962640864} a^{6} + \frac{252098006607999064878983}{8474179180675233962640864} a^{5} - \frac{205283098311117601793149}{2118544795168808490660216} a^{4} - \frac{499560902276027388612961}{4237089590337616981320432} a^{3} - \frac{322816753890040526996051}{1059272397584404245330108} a^{2} + \frac{2117189050939163355332519}{6355634385506425471980648} a + \frac{1053714589184492051748949}{3177817192753212735990324}$

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:  \( 415749575255 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$A_{11}$ (as 11T7):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 19958400
The 31 conjugacy class representatives for $A_{11}$
Character table for $A_{11}$ 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 ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ ${\href{/LocalNumberField/7.11.0.1}{11} }$ R ${\href{/LocalNumberField/13.7.0.1}{7} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.11.0.1}{11} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.7.0.1}{7} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.11.0.1}{11} }$ ${\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.7.0.1}{7} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.11.0.1}{11} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.4.6.9$x^{4} + 2 x^{3} + 6$$4$$1$$6$$D_{4}$$[2, 2]^{2}$
$3$$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
3.3.3.2$x^{3} + 3 x + 3$$3$$1$$3$$S_3$$[3/2]_{2}$
3.6.9.12$x^{6} + 3 x^{4} + 3$$6$$1$$9$$D_{6}$$[2]_{2}^{2}$
$11$11.11.18.4$x^{11} + 88 x^{8} + 11$$11$$1$$18$$C_{11}:C_5$$[9/5]_{5}$