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Properties

Label 20.4.57722337794...4089.6
Degree $20$
Signature $[4, 8]$
Discriminant $11^{16}\cdot 23^{4}\cdot 67^{2}$
Root discriminant $19.41$
Ramified primes $11, 23, 67$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T341

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 1, -12, -17, 63, 92, -141, -314, 171, 496, -121, -496, 171, 314, -141, -92, 63, 17, -12, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 - 12*x^18 + 17*x^17 + 63*x^16 - 92*x^15 - 141*x^14 + 314*x^13 + 171*x^12 - 496*x^11 - 121*x^10 + 496*x^9 + 171*x^8 - 314*x^7 - 141*x^6 + 92*x^5 + 63*x^4 - 17*x^3 - 12*x^2 + x + 1)
 
gp: K = bnfinit(x^20 - x^19 - 12*x^18 + 17*x^17 + 63*x^16 - 92*x^15 - 141*x^14 + 314*x^13 + 171*x^12 - 496*x^11 - 121*x^10 + 496*x^9 + 171*x^8 - 314*x^7 - 141*x^6 + 92*x^5 + 63*x^4 - 17*x^3 - 12*x^2 + x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} - 12 x^{18} + 17 x^{17} + 63 x^{16} - 92 x^{15} - 141 x^{14} + 314 x^{13} + 171 x^{12} - 496 x^{11} - 121 x^{10} + 496 x^{9} + 171 x^{8} - 314 x^{7} - 141 x^{6} + 92 x^{5} + 63 x^{4} - 17 x^{3} - 12 x^{2} + x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(57722337794481266110634089=11^{16}\cdot 23^{4}\cdot 67^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $19.41$
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:  $11, 23, 67$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{89} a^{17} + \frac{5}{89} a^{16} - \frac{8}{89} a^{15} - \frac{17}{89} a^{14} - \frac{1}{89} a^{13} - \frac{7}{89} a^{12} - \frac{24}{89} a^{11} + \frac{30}{89} a^{10} - \frac{33}{89} a^{9} - \frac{35}{89} a^{8} + \frac{41}{89} a^{7} + \frac{15}{89} a^{6} + \frac{29}{89} a^{5} + \frac{34}{89} a^{4} - \frac{44}{89} a^{3} + \frac{5}{89} a^{2} - \frac{8}{89} a - \frac{34}{89}$, $\frac{1}{513119443} a^{18} + \frac{1104819}{513119443} a^{17} + \frac{220538183}{513119443} a^{16} + \frac{15619027}{513119443} a^{15} - \frac{147669859}{513119443} a^{14} - \frac{207091049}{513119443} a^{13} - \frac{14027713}{513119443} a^{12} + \frac{69996717}{513119443} a^{11} + \frac{224616406}{513119443} a^{10} - \frac{59118386}{513119443} a^{9} - \frac{207320245}{513119443} a^{8} - \frac{154853376}{513119443} a^{7} + \frac{88977744}{513119443} a^{6} + \frac{202251428}{513119443} a^{5} - \frac{13760977}{513119443} a^{4} - \frac{65096391}{513119443} a^{3} - \frac{255130505}{513119443} a^{2} + \frac{237485686}{513119443} a - \frac{190257772}{513119443}$, $\frac{1}{513119443} a^{19} + \frac{1104808}{513119443} a^{17} + \frac{863238}{22309541} a^{16} + \frac{253453434}{513119443} a^{15} - \frac{57100893}{513119443} a^{14} - \frac{3072442}{513119443} a^{13} - \frac{19329903}{513119443} a^{12} - \frac{70869339}{513119443} a^{11} + \frac{6343277}{513119443} a^{10} - \frac{243844578}{513119443} a^{9} + \frac{235150534}{513119443} a^{8} + \frac{29872866}{513119443} a^{7} - \frac{129295203}{513119443} a^{6} - \frac{170001929}{513119443} a^{5} - \frac{828843}{22309541} a^{4} + \frac{244004367}{513119443} a^{3} - \frac{164561464}{513119443} a^{2} - \frac{348670}{513119443} a + \frac{122177946}{513119443}$

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:  $11$
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:  \( 70471.9243098 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T341:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 5120
The 224 conjugacy class representatives for t20n341 are not computed
Character table for t20n341 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.8.7597521819283.2, 10.2.330327035621.1, 10.4.4930254263.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 20 siblings: 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
$11$11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
$23$23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$67$$\Q_{67}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{67}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{67}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{67}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{67}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{67}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{67}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{67}$$x + 4$$1$$1$$0$Trivial$[\ ]$
67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.4.2.1$x^{4} + 1541 x^{2} + 646416$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$