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Properties

Label 20.20.1954417180...0272.1
Degree $20$
Signature $[20, 0]$
Discriminant $2^{16}\cdot 13^{15}\cdot 17^{12}$
Root discriminant $65.25$
Ramified primes $2, 13, 17$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $S_5$ (as 20T35)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![89557, 0, -461747, 0, 965263, 0, -1076170, 0, 708441, 0, -288141, 0, 73285, 0, -11466, 0, 1051, 0, -51, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 51*x^18 + 1051*x^16 - 11466*x^14 + 73285*x^12 - 288141*x^10 + 708441*x^8 - 1076170*x^6 + 965263*x^4 - 461747*x^2 + 89557)
 
gp: K = bnfinit(x^20 - 51*x^18 + 1051*x^16 - 11466*x^14 + 73285*x^12 - 288141*x^10 + 708441*x^8 - 1076170*x^6 + 965263*x^4 - 461747*x^2 + 89557, 1)
 

Normalized defining polynomial

\( x^{20} - 51 x^{18} + 1051 x^{16} - 11466 x^{14} + 73285 x^{12} - 288141 x^{10} + 708441 x^{8} - 1076170 x^{6} + 965263 x^{4} - 461747 x^{2} + 89557 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[20, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1954417180834054231108741720904630272=2^{16}\cdot 13^{15}\cdot 17^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $65.25$
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:  $2, 13, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{6} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{4} - \frac{1}{2} a^{2} - \frac{1}{4}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{5} - \frac{1}{2} a^{3} - \frac{1}{4} a$, $\frac{1}{8} a^{10} - \frac{1}{8} a^{9} - \frac{1}{8} a^{8} - \frac{1}{4} a^{7} + \frac{1}{8} a^{6} + \frac{1}{8} a^{5} + \frac{1}{8} a^{4} + \frac{1}{4} a^{3} + \frac{3}{8} a^{2} + \frac{3}{8} a + \frac{1}{8}$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{8} - \frac{1}{8} a^{7} - \frac{1}{4} a^{6} + \frac{1}{8} a^{4} + \frac{1}{8} a^{3} + \frac{1}{4} a^{2} + \frac{1}{4} a + \frac{3}{8}$, $\frac{1}{16} a^{12} - \frac{1}{16} a^{10} - \frac{1}{8} a^{8} - \frac{1}{16} a^{6} - \frac{1}{4} a^{5} - \frac{1}{8} a^{4} + \frac{1}{4} a^{3} + \frac{7}{16} a^{2} + \frac{1}{4} a - \frac{3}{16}$, $\frac{1}{16} a^{13} - \frac{1}{16} a^{11} - \frac{1}{8} a^{9} - \frac{1}{16} a^{7} - \frac{1}{4} a^{6} - \frac{1}{8} a^{5} - \frac{1}{4} a^{4} + \frac{7}{16} a^{3} - \frac{1}{4} a^{2} - \frac{3}{16} a - \frac{1}{2}$, $\frac{1}{16} a^{14} - \frac{1}{16} a^{10} - \frac{1}{8} a^{9} - \frac{1}{16} a^{8} - \frac{1}{16} a^{6} + \frac{1}{8} a^{5} + \frac{3}{16} a^{4} - \frac{1}{4} a^{3} + \frac{1}{8} a^{2} + \frac{1}{8} a - \frac{5}{16}$, $\frac{1}{16} a^{15} - \frac{1}{16} a^{11} + \frac{1}{16} a^{9} - \frac{1}{8} a^{8} + \frac{3}{16} a^{7} - \frac{1}{4} a^{6} + \frac{1}{16} a^{5} - \frac{1}{8} a^{4} - \frac{1}{8} a^{3} - \frac{1}{2} a^{2} + \frac{5}{16} a - \frac{3}{8}$, $\frac{1}{32} a^{16} - \frac{1}{16} a^{10} + \frac{3}{32} a^{8} - \frac{1}{4} a^{7} + \frac{3}{16} a^{6} - \frac{1}{4} a^{5} + \frac{1}{16} a^{4} + \frac{1}{4} a^{3} + \frac{7}{16} a^{2} - \frac{1}{2} a - \frac{5}{32}$, $\frac{1}{32} a^{17} - \frac{1}{16} a^{11} + \frac{3}{32} a^{9} + \frac{3}{16} a^{7} - \frac{1}{4} a^{6} + \frac{1}{16} a^{5} + \frac{7}{16} a^{3} - \frac{5}{32} a - \frac{1}{4}$, $\frac{1}{191101472} a^{18} - \frac{408359}{191101472} a^{16} - \frac{2879309}{95550736} a^{14} + \frac{581537}{95550736} a^{12} + \frac{6756255}{191101472} a^{10} - \frac{1}{8} a^{9} - \frac{11843309}{191101472} a^{8} + \frac{22534339}{95550736} a^{6} - \frac{1}{8} a^{5} - \frac{797955}{95550736} a^{4} + \frac{75260601}{191101472} a^{2} - \frac{1}{8} a + \frac{88516425}{191101472}$, $\frac{1}{15861422176} a^{19} + \frac{190693113}{15861422176} a^{17} - \frac{28313299}{3965355544} a^{15} - \frac{6308820}{495669443} a^{13} + \frac{986151299}{15861422176} a^{11} - \frac{1122620615}{15861422176} a^{9} - \frac{1016579915}{7930711088} a^{7} - \frac{216681625}{1982677772} a^{5} - \frac{3949814153}{15861422176} a^{3} - \frac{1}{2} a^{2} - \frac{2300251975}{15861422176} a$

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:  $19$
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:  \( 964006871003 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_5$ (as 20T35):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 120
The 7 conjugacy class representatives for $S_5$
Character table for $S_5$

Intermediate fields

10.10.387736744636204288.1

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

Sibling fields

Degree 5 sibling: data not computed
Degree 6 sibling: data not computed
Degree 10 siblings: data not computed
Degree 12 sibling: data not computed
Degree 15 sibling: data not computed
Degree 20 siblings: data not computed
Degree 24 sibling: data not computed
Degree 30 siblings: data not computed
Degree 40 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 ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{5}$ R R ${\href{/LocalNumberField/19.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
2Data not computed
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.6.5.3$x^{6} - 208$$6$$1$$5$$C_6$$[\ ]_{6}$
13.6.5.3$x^{6} - 208$$6$$1$$5$$C_6$$[\ ]_{6}$
13.6.5.3$x^{6} - 208$$6$$1$$5$$C_6$$[\ ]_{6}$
$17$17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.6.4.1$x^{6} + 136 x^{3} + 7803$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
17.6.4.1$x^{6} + 136 x^{3} + 7803$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
17.6.4.1$x^{6} + 136 x^{3} + 7803$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$