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Properties

Label 11.11.4487278306...5625.1
Degree $11$
Signature $[11, 0]$
Discriminant $5^{10}\cdot 11^{16}$
Root discriminant $141.31$
Ramified primes $5, 11$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_{11}:C_5$ (as 11T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-12675, -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 - 12675)
 
gp: K = bnfinit(x^11 - 55*x^9 + 1100*x^7 - 9625*x^5 + 34375*x^3 - 34375*x - 12675, 1)
 

Normalized defining polynomial

\( x^{11} - 55 x^{9} + 1100 x^{7} - 9625 x^{5} + 34375 x^{3} - 34375 x - 12675 \)

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:  \(448727830698946884765625=5^{10}\cdot 11^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $141.31$
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:  $5, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{355} a^{6} - \frac{25}{71} a^{5} - \frac{6}{71} a^{4} - \frac{14}{71} a^{3} - \frac{26}{71} a^{2} - \frac{1}{71} a + \frac{21}{71}$, $\frac{1}{355} a^{7} - \frac{7}{71} a^{5} + \frac{17}{71} a^{4} - \frac{1}{71} a^{3} + \frac{15}{71} a^{2} - \frac{33}{71} a - \frac{2}{71}$, $\frac{1}{355} a^{8} - \frac{6}{71} a^{5} + \frac{2}{71} a^{4} + \frac{22}{71} a^{3} - \frac{20}{71} a^{2} + \frac{34}{71} a + \frac{25}{71}$, $\frac{1}{355} a^{9} + \frac{33}{71} a^{5} - \frac{16}{71} a^{4} - \frac{14}{71} a^{3} + \frac{35}{71} a^{2} - \frac{5}{71} a - \frac{9}{71}$, $\frac{1}{355} a^{10} - \frac{9}{71} a^{5} - \frac{18}{71} a^{4} + \frac{2}{71} a^{3} + \frac{25}{71} a^{2} + \frac{14}{71} a + \frac{14}{71}$

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:  \( 677575079.592 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{11}:C_5$ (as 11T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 55
The 7 conjugacy class representatives for $C_{11}:C_5$
Character table for $C_{11}:C_5$

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 ${\href{/LocalNumberField/2.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }$ ${\href{/LocalNumberField/3.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ R ${\href{/LocalNumberField/7.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ R ${\href{/LocalNumberField/13.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.11.0.1}{11} }$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.11.0.1}{11} }$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
$5$5.11.10.1$x^{11} - 5$$11$$1$$10$$C_{11}:C_5$$[\ ]_{11}^{5}$
$11$11.11.16.4$x^{11} + 22 x^{6} + 11$$11$$1$$16$$C_{11}:C_5$$[8/5]_{5}$