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Properties

Label 20.12.2118863682...5625.1
Degree $20$
Signature $[12, 4]$
Discriminant $3^{16}\cdot 5^{14}\cdot 73^{8}$
Root discriminant $41.33$
Ramified primes $3, 5, 73$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T230

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, -21, 0, 68, 0, 42, 0, -44, 0, -117, 0, -44, 0, 42, 0, 68, 0, -21, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 21*x^18 + 68*x^16 + 42*x^14 - 44*x^12 - 117*x^10 - 44*x^8 + 42*x^6 + 68*x^4 - 21*x^2 + 1)
 
gp: K = bnfinit(x^20 - 21*x^18 + 68*x^16 + 42*x^14 - 44*x^12 - 117*x^10 - 44*x^8 + 42*x^6 + 68*x^4 - 21*x^2 + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 21 x^{18} + 68 x^{16} + 42 x^{14} - 44 x^{12} - 117 x^{10} - 44 x^{8} + 42 x^{6} + 68 x^{4} - 21 x^{2} + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(211886368245842690175787353515625=3^{16}\cdot 5^{14}\cdot 73^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $41.33$
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:  $3, 5, 73$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{4} + \frac{1}{3}$, $\frac{1}{3} a^{9} - \frac{1}{3} a^{5} + \frac{1}{3} a$, $\frac{1}{30} a^{10} - \frac{1}{6} a^{9} + \frac{1}{15} a^{8} + \frac{1}{15} a^{6} + \frac{1}{6} a^{5} - \frac{4}{15} a^{4} - \frac{4}{15} a^{2} - \frac{1}{6} a + \frac{11}{30}$, $\frac{1}{30} a^{11} - \frac{1}{10} a^{9} + \frac{1}{15} a^{7} - \frac{1}{2} a^{6} - \frac{1}{10} a^{5} - \frac{4}{15} a^{3} - \frac{1}{2} a^{2} + \frac{1}{5} a - \frac{1}{2}$, $\frac{1}{90} a^{12} - \frac{1}{6} a^{9} - \frac{2}{15} a^{8} - \frac{1}{2} a^{7} - \frac{3}{10} a^{6} + \frac{1}{6} a^{5} + \frac{1}{5} a^{4} - \frac{1}{2} a^{3} - \frac{1}{5} a^{2} + \frac{1}{3} a + \frac{13}{90}$, $\frac{1}{90} a^{13} + \frac{1}{30} a^{9} - \frac{1}{6} a^{8} - \frac{3}{10} a^{7} - \frac{1}{2} a^{6} + \frac{1}{30} a^{5} + \frac{1}{6} a^{4} - \frac{1}{5} a^{3} + \frac{14}{45} a - \frac{1}{6}$, $\frac{1}{90} a^{14} - \frac{1}{30} a^{8} - \frac{1}{2} a^{7} - \frac{1}{30} a^{6} - \frac{4}{15} a^{4} - \frac{19}{45} a^{2} - \frac{1}{30}$, $\frac{1}{90} a^{15} - \frac{1}{30} a^{9} - \frac{1}{6} a^{8} - \frac{1}{30} a^{7} - \frac{4}{15} a^{5} - \frac{1}{3} a^{4} - \frac{19}{45} a^{3} - \frac{1}{30} a + \frac{1}{3}$, $\frac{1}{270} a^{16} + \frac{1}{270} a^{12} - \frac{1}{6} a^{9} - \frac{13}{90} a^{8} - \frac{1}{2} a^{7} - \frac{1}{6} a^{6} - \frac{1}{3} a^{5} - \frac{52}{135} a^{4} - \frac{1}{2} a^{3} - \frac{1}{6} a^{2} - \frac{1}{6} a + \frac{8}{135}$, $\frac{1}{270} a^{17} + \frac{1}{270} a^{13} + \frac{1}{45} a^{9} - \frac{1}{6} a^{8} - \frac{1}{6} a^{7} + \frac{121}{270} a^{5} + \frac{1}{6} a^{4} - \frac{1}{6} a^{3} - \frac{1}{2} a^{2} + \frac{61}{270} a - \frac{1}{6}$, $\frac{1}{270} a^{18} + \frac{1}{270} a^{14} - \frac{1}{90} a^{10} + \frac{1}{10} a^{8} + \frac{103}{270} a^{6} - \frac{7}{30} a^{4} - \frac{1}{2} a^{3} + \frac{133}{270} a^{2} - \frac{1}{30}$, $\frac{1}{270} a^{19} + \frac{1}{270} a^{15} - \frac{1}{90} a^{11} + \frac{1}{10} a^{9} + \frac{103}{270} a^{7} - \frac{7}{30} a^{5} - \frac{1}{2} a^{4} + \frac{133}{270} a^{3} - \frac{1}{30} a$

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:  $15$
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:  \( 1874267335.76 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T230:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 1920
The 24 conjugacy class representatives for t20n230
Character table for t20n230 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.10791225.1, 10.6.14556317125078125.1, 10.10.582252685003125.1, 10.6.2911263425015625.1

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

Sibling fields

Degree 10 sibling: data not computed
Degree 20 siblings: data not computed
Degree 30 siblings: data not computed
Degree 32 sibling: data not computed
Degree 40 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ R R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.6.6.3$x^{6} + 3 x^{4} + 9$$3$$2$$6$$D_{6}$$[3/2]_{2}^{2}$
3.6.6.3$x^{6} + 3 x^{4} + 9$$3$$2$$6$$D_{6}$$[3/2]_{2}^{2}$
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$73$73.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
73.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
73.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
73.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
73.6.4.2$x^{6} - 73 x^{3} + 58619$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
73.6.4.2$x^{6} - 73 x^{3} + 58619$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$