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Properties

Label 11.11.2853116706...0000.1
Degree $11$
Signature $[11, 0]$
Discriminant $2^{10}\cdot 5^{10}\cdot 11^{11}$
Root discriminant $89.22$
Ramified primes $2, 5, 11$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $F_{11}$ (as 11T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-13900, -34375, 0, 34375, 0, -9625, 0, 1100, 0, -55, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^11 - 55*x^9 + 1100*x^7 - 9625*x^5 + 34375*x^3 - 34375*x - 13900)
 
gp: K = bnfinit(x^11 - 55*x^9 + 1100*x^7 - 9625*x^5 + 34375*x^3 - 34375*x - 13900, 1)
 

Normalized defining polynomial

\( x^{11} - 55 x^{9} + 1100 x^{7} - 9625 x^{5} + 34375 x^{3} - 34375 x - 13900 \)

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

Invariants

Degree:  $11$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[11, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(2853116706110000000000=2^{10}\cdot 5^{10}\cdot 11^{11}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $89.22$
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:  $2, 5, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{145} a^{6} + \frac{8}{29} a^{5} - \frac{6}{29} a^{4} + \frac{3}{29} a^{3} - \frac{13}{29} a^{2} + \frac{14}{29} a + \frac{8}{29}$, $\frac{1}{145} a^{7} - \frac{7}{29} a^{5} + \frac{11}{29} a^{4} + \frac{12}{29} a^{3} + \frac{12}{29} a^{2} - \frac{1}{29} a - \frac{1}{29}$, $\frac{1}{145} a^{8} + \frac{1}{29} a^{5} + \frac{5}{29} a^{4} + \frac{1}{29} a^{3} + \frac{8}{29} a^{2} - \frac{4}{29} a - \frac{10}{29}$, $\frac{1}{145} a^{9} - \frac{6}{29} a^{5} + \frac{2}{29} a^{4} - \frac{7}{29} a^{3} + \frac{3}{29} a^{2} + \frac{7}{29} a - \frac{11}{29}$, $\frac{1}{145} a^{10} + \frac{10}{29} a^{5} - \frac{13}{29} a^{4} + \frac{6}{29} a^{3} - \frac{6}{29} a^{2} + \frac{3}{29} a + \frac{8}{29}$

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:  $10$
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:  \( 52622964.70901569 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$F_{11}$ (as 11T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 110
The 11 conjugacy class representatives for $F_{11}$
Character table for $F_{11}$

Intermediate fields

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

Sibling fields

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 ${\href{/LocalNumberField/3.10.0.1}{10} }{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ R ${\href{/LocalNumberField/7.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ R ${\href{/LocalNumberField/13.10.0.1}{10} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.11.0.1}{11} }$ ${\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$$0$Trivial$[\ ]$
2.10.10.11$x^{10} - x^{8} + 3 x^{6} + x^{2} - 3$$2$$5$$10$$C_{10}$$[2]^{5}$
$5$5.11.10.1$x^{11} - 5$$11$$1$$10$$C_{11}:C_5$$[\ ]_{11}^{5}$
$11$11.11.11.1$x^{11} + 110 x + 11$$11$$1$$11$$F_{11}$$[11/10]_{10}$