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Properties

Label 20.2.30988918815...0000.1
Degree $20$
Signature $[2, 9]$
Discriminant $-\,2^{4}\cdot 5^{14}\cdot 19^{4}\cdot 29^{7}\cdot 109^{4}$
Root discriminant $53.03$
Ramified primes $2, 5, 19, 29, 109$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group 20T955

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-725, 0, -14065, 0, -63104, 0, -117044, 0, -103315, 0, -45946, 0, -10001, 0, -696, 0, 105, 0, 21, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 21*x^18 + 105*x^16 - 696*x^14 - 10001*x^12 - 45946*x^10 - 103315*x^8 - 117044*x^6 - 63104*x^4 - 14065*x^2 - 725)
 
gp: K = bnfinit(x^20 + 21*x^18 + 105*x^16 - 696*x^14 - 10001*x^12 - 45946*x^10 - 103315*x^8 - 117044*x^6 - 63104*x^4 - 14065*x^2 - 725, 1)
 

Normalized defining polynomial

\( x^{20} + 21 x^{18} + 105 x^{16} - 696 x^{14} - 10001 x^{12} - 45946 x^{10} - 103315 x^{8} - 117044 x^{6} - 63104 x^{4} - 14065 x^{2} - 725 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-30988918815000698609123925781250000=-\,2^{4}\cdot 5^{14}\cdot 19^{4}\cdot 29^{7}\cdot 109^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $53.03$
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:  $2, 5, 19, 29, 109$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{5} a^{10} - \frac{2}{5} a^{8} - \frac{2}{5} a^{6} - \frac{2}{5} a^{4} + \frac{1}{5} a^{2}$, $\frac{1}{5} a^{11} - \frac{2}{5} a^{9} - \frac{2}{5} a^{7} - \frac{2}{5} a^{5} + \frac{1}{5} a^{3}$, $\frac{1}{10} a^{12} - \frac{1}{10} a^{10} - \frac{2}{5} a^{8} - \frac{1}{2} a^{7} - \frac{2}{5} a^{6} - \frac{1}{2} a^{5} - \frac{1}{10} a^{4} - \frac{1}{2} a^{3} - \frac{2}{5} a^{2} - \frac{1}{2}$, $\frac{1}{10} a^{13} - \frac{1}{10} a^{11} - \frac{2}{5} a^{9} - \frac{1}{2} a^{8} - \frac{2}{5} a^{7} - \frac{1}{2} a^{6} - \frac{1}{10} a^{5} - \frac{1}{2} a^{4} - \frac{2}{5} a^{3} - \frac{1}{2} a$, $\frac{1}{10} a^{14} - \frac{1}{10} a^{10} - \frac{1}{2} a^{9} + \frac{2}{5} a^{8} - \frac{3}{10} a^{6} - \frac{3}{10} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{10} a^{15} - \frac{1}{10} a^{11} - \frac{1}{10} a^{10} + \frac{2}{5} a^{9} + \frac{1}{5} a^{8} - \frac{3}{10} a^{7} + \frac{1}{5} a^{6} - \frac{3}{10} a^{5} - \frac{3}{10} a^{4} - \frac{1}{2} a^{3} + \frac{2}{5} a^{2} - \frac{1}{2} a$, $\frac{1}{10} a^{16} - \frac{1}{10} a^{11} - \frac{1}{10} a^{10} + \frac{1}{5} a^{9} + \frac{1}{10} a^{8} - \frac{3}{10} a^{7} + \frac{1}{10} a^{6} + \frac{1}{5} a^{5} + \frac{1}{5} a^{4} - \frac{1}{10} a^{3} - \frac{3}{10} a^{2} - \frac{1}{2}$, $\frac{1}{10} a^{17} - \frac{1}{10} a^{11} - \frac{1}{10} a^{10} + \frac{1}{10} a^{9} - \frac{3}{10} a^{8} - \frac{2}{5} a^{7} + \frac{1}{5} a^{6} - \frac{3}{10} a^{5} + \frac{1}{5} a^{4} + \frac{1}{5} a^{3} + \frac{2}{5} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2519901428650} a^{18} - \frac{508702312}{1259950714325} a^{16} - \frac{1341229646}{50398028573} a^{14} - \frac{4427912818}{1259950714325} a^{12} - \frac{1}{10} a^{11} - \frac{35390160823}{1259950714325} a^{10} - \frac{3}{10} a^{9} + \frac{480987087782}{1259950714325} a^{8} - \frac{3}{10} a^{7} - \frac{234314318951}{503980285730} a^{6} - \frac{3}{10} a^{5} - \frac{844620150999}{2519901428650} a^{4} - \frac{1}{10} a^{3} + \frac{82353721841}{2519901428650} a^{2} - \frac{1}{2} a - \frac{246515742439}{503980285730}$, $\frac{1}{12599507143250} a^{19} + \frac{250972738241}{12599507143250} a^{17} - \frac{1341229646}{251990142865} a^{15} + \frac{243134317229}{12599507143250} a^{13} - \frac{35390160823}{6299753571625} a^{11} - \frac{1}{10} a^{10} - \frac{5841759681791}{12599507143250} a^{9} - \frac{3}{10} a^{8} - \frac{545540402346}{1259950714325} a^{7} + \frac{1}{5} a^{6} - \frac{800295289797}{6299753571625} a^{5} - \frac{3}{10} a^{4} - \frac{1848749210567}{6299753571625} a^{3} - \frac{1}{10} a^{2} + \frac{257464543291}{2519901428650} a$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{3}$, which has order $3$ (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:  \( 1042391985.69 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T955:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 819200
The 275 conjugacy class representatives for t20n955 are not computed
Character table for t20n955 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.10.8172298511640625.1

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
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 R ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }$ R $20$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ $16{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$2$2.4.4.4$x^{4} - 5$$2$$2$$4$$D_{4}$$[2, 2]^{2}$
2.8.0.1$x^{8} + x^{4} + x^{3} + x + 1$$1$$8$$0$$C_8$$[\ ]^{8}$
2.8.0.1$x^{8} + x^{4} + x^{3} + x + 1$$1$$8$$0$$C_8$$[\ ]^{8}$
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$19$19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.4.2.2$x^{4} - 19 x^{2} + 722$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
19.4.2.2$x^{4} - 19 x^{2} + 722$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
19.10.0.1$x^{10} + x^{2} - 2 x + 14$$1$$10$$0$$C_{10}$$[\ ]^{10}$
$29$$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.8.7.4$x^{8} + 232$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
$109$109.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
109.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
109.2.1.2$x^{2} + 654$$2$$1$$1$$C_2$$[\ ]_{2}$
109.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
109.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
109.2.1.2$x^{2} + 654$$2$$1$$1$$C_2$$[\ ]_{2}$
109.2.1.2$x^{2} + 654$$2$$1$$1$$C_2$$[\ ]_{2}$
109.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
109.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
109.2.1.2$x^{2} + 654$$2$$1$$1$$C_2$$[\ ]_{2}$