↑

Properties

Label 11.11.1937634505...0000.1
Degree $11$
Signature $[11, 0]$
Discriminant $2^{10}\cdot 5^{5}\cdot 7^{10}\cdot 11^{8}$
Root discriminant $130.93$
Ramified primes $2, 5, 7, 11$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $F_{11}$ (as 11T4)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3279, 19909, 36219, 20673, -5784, -7812, -28, 996, 31, -53, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^11 - x^10 - 53*x^9 + 31*x^8 + 996*x^7 - 28*x^6 - 7812*x^5 - 5784*x^4 + 20673*x^3 + 36219*x^2 + 19909*x + 3279)
 
gp: K = bnfinit(x^11 - x^10 - 53*x^9 + 31*x^8 + 996*x^7 - 28*x^6 - 7812*x^5 - 5784*x^4 + 20673*x^3 + 36219*x^2 + 19909*x + 3279, 1)
 

Normalized defining polynomial

\( x^{11} - x^{10} - 53 x^{9} + 31 x^{8} + 996 x^{7} - 28 x^{6} - 7812 x^{5} - 5784 x^{4} + 20673 x^{3} + 36219 x^{2} + 19909 x + 3279 \)

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:  \(193763450514676380800000=2^{10}\cdot 5^{5}\cdot 7^{10}\cdot 11^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $130.93$
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:  $2, 5, 7, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{10858380347} a^{10} - \frac{462589751}{10858380347} a^{9} + \frac{5122261873}{10858380347} a^{8} - \frac{4502898877}{10858380347} a^{7} + \frac{3059186771}{10858380347} a^{6} - \frac{1455762823}{10858380347} a^{5} - \frac{685059024}{10858380347} a^{4} - \frac{4860779434}{10858380347} a^{3} - \frac{2890385116}{10858380347} a^{2} + \frac{92665252}{10858380347} a + \frac{1524436607}{10858380347}$

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:  \( 501847260.162 \) (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 R R ${\href{/LocalNumberField/13.10.0.1}{10} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
$2$2.11.10.1$x^{11} - 2$$11$$1$$10$$F_{11}$$[\ ]_{11}^{10}$
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$7$7.11.10.1$x^{11} - 7$$11$$1$$10$$F_{11}$$[\ ]_{11}^{10}$
$11$$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.5.2t1.1c1$1$ $ 5 $ $x^{2} - x - 1$ $C_2$ (as 2T1) $1$ $1$
1.11.5t1.1c1$1$ $ 11 $ $x^{5} - x^{4} - 4 x^{3} + 3 x^{2} + 3 x - 1$ $C_5$ (as 5T1) $0$ $1$
1.5_11.10t1.1c1$1$ $ 5 \cdot 11 $ $x^{10} - x^{9} - 13 x^{8} + 8 x^{7} + 46 x^{6} - 11 x^{5} - 52 x^{4} + 7 x^{3} + 18 x^{2} - 3 x - 1$ $C_{10}$ (as 10T1) $0$ $1$
1.11.5t1.1c2$1$ $ 11 $ $x^{5} - x^{4} - 4 x^{3} + 3 x^{2} + 3 x - 1$ $C_5$ (as 5T1) $0$ $1$
1.11.5t1.1c3$1$ $ 11 $ $x^{5} - x^{4} - 4 x^{3} + 3 x^{2} + 3 x - 1$ $C_5$ (as 5T1) $0$ $1$
1.11.5t1.1c4$1$ $ 11 $ $x^{5} - x^{4} - 4 x^{3} + 3 x^{2} + 3 x - 1$ $C_5$ (as 5T1) $0$ $1$
1.5_11.10t1.1c2$1$ $ 5 \cdot 11 $ $x^{10} - x^{9} - 13 x^{8} + 8 x^{7} + 46 x^{6} - 11 x^{5} - 52 x^{4} + 7 x^{3} + 18 x^{2} - 3 x - 1$ $C_{10}$ (as 10T1) $0$ $1$
1.5_11.10t1.1c3$1$ $ 5 \cdot 11 $ $x^{10} - x^{9} - 13 x^{8} + 8 x^{7} + 46 x^{6} - 11 x^{5} - 52 x^{4} + 7 x^{3} + 18 x^{2} - 3 x - 1$ $C_{10}$ (as 10T1) $0$ $1$
1.5_11.10t1.1c4$1$ $ 5 \cdot 11 $ $x^{10} - x^{9} - 13 x^{8} + 8 x^{7} + 46 x^{6} - 11 x^{5} - 52 x^{4} + 7 x^{3} + 18 x^{2} - 3 x - 1$ $C_{10}$ (as 10T1) $0$ $1$
* 10.2e10_5e5_7e10_11e8.11t4.1c1$10$ $ 2^{10} \cdot 5^{5} \cdot 7^{10} \cdot 11^{8}$ $x^{11} - x^{10} - 53 x^{9} + 31 x^{8} + 996 x^{7} - 28 x^{6} - 7812 x^{5} - 5784 x^{4} + 20673 x^{3} + 36219 x^{2} + 19909 x + 3279$ $F_{11}$ (as 11T4) $1$ $10$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.