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Properties

Label 20.12.2209110141...0000.1
Degree $20$
Signature $[12, 4]$
Discriminant $2^{16}\cdot 3^{4}\cdot 5^{10}\cdot 17^{4}\cdot 29^{4}\cdot 721381$
Root discriminant $32.90$
Ramified primes $2, 3, 5, 17, 29, 721381$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1036

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -3, 30, -14, -98, 225, -288, -202, 1127, -668, -1170, 1414, 280, -1044, 278, 304, -191, -3, 32, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 32*x^18 - 3*x^17 - 191*x^16 + 304*x^15 + 278*x^14 - 1044*x^13 + 280*x^12 + 1414*x^11 - 1170*x^10 - 668*x^9 + 1127*x^8 - 202*x^7 - 288*x^6 + 225*x^5 - 98*x^4 - 14*x^3 + 30*x^2 - 3*x - 1)
 
gp: K = bnfinit(x^20 - 10*x^19 + 32*x^18 - 3*x^17 - 191*x^16 + 304*x^15 + 278*x^14 - 1044*x^13 + 280*x^12 + 1414*x^11 - 1170*x^10 - 668*x^9 + 1127*x^8 - 202*x^7 - 288*x^6 + 225*x^5 - 98*x^4 - 14*x^3 + 30*x^2 - 3*x - 1, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 32 x^{18} - 3 x^{17} - 191 x^{16} + 304 x^{15} + 278 x^{14} - 1044 x^{13} + 280 x^{12} + 1414 x^{11} - 1170 x^{10} - 668 x^{9} + 1127 x^{8} - 202 x^{7} - 288 x^{6} + 225 x^{5} - 98 x^{4} - 14 x^{3} + 30 x^{2} - 3 x - 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(2209110141965921447040000000000=2^{16}\cdot 3^{4}\cdot 5^{10}\cdot 17^{4}\cdot 29^{4}\cdot 721381\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $32.90$
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:  $2, 3, 5, 17, 29, 721381$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{18} a^{16} - \frac{4}{9} a^{15} - \frac{2}{9} a^{14} + \frac{1}{3} a^{13} + \frac{1}{3} a^{12} + \frac{4}{9} a^{11} - \frac{1}{9} a^{10} - \frac{1}{9} a^{9} - \frac{1}{3} a^{8} + \frac{2}{9} a^{7} + \frac{4}{9} a^{6} + \frac{4}{9} a^{5} + \frac{1}{6} a^{4} - \frac{2}{9} a^{3} - \frac{1}{9} a^{2} + \frac{1}{9} a - \frac{7}{18}$, $\frac{1}{18} a^{17} + \frac{2}{9} a^{15} - \frac{4}{9} a^{14} + \frac{1}{9} a^{12} + \frac{4}{9} a^{11} - \frac{2}{9} a^{9} - \frac{4}{9} a^{8} + \frac{2}{9} a^{7} - \frac{5}{18} a^{5} + \frac{1}{9} a^{4} + \frac{1}{9} a^{3} + \frac{2}{9} a^{2} - \frac{1}{2} a - \frac{1}{9}$, $\frac{1}{54} a^{18} + \frac{4}{9} a^{15} - \frac{1}{27} a^{14} - \frac{11}{27} a^{13} + \frac{1}{27} a^{12} + \frac{11}{27} a^{11} + \frac{11}{27} a^{10} - \frac{1}{3} a^{9} - \frac{4}{27} a^{8} - \frac{8}{27} a^{7} + \frac{17}{54} a^{6} + \frac{1}{9} a^{5} + \frac{4}{27} a^{4} + \frac{1}{27} a^{3} - \frac{1}{54} a^{2} + \frac{4}{27} a - \frac{13}{27}$, $\frac{1}{54} a^{19} - \frac{13}{27} a^{15} + \frac{10}{27} a^{14} + \frac{10}{27} a^{13} - \frac{7}{27} a^{12} - \frac{4}{27} a^{11} - \frac{4}{9} a^{10} - \frac{7}{27} a^{9} + \frac{10}{27} a^{8} - \frac{25}{54} a^{7} - \frac{4}{9} a^{6} - \frac{11}{27} a^{5} - \frac{8}{27} a^{4} - \frac{13}{54} a^{3} + \frac{1}{27} a^{2} - \frac{10}{27} a + \frac{1}{9}$

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:  \( 110134185.852 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1036:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 14745600
The 396 conjugacy class representatives for t20n1036 are not computed
Character table for t20n1036 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.6.1749952800000.1

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 R R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ $20$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$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}$
3.6.0.1$x^{6} - x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
3.6.0.1$x^{6} - x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
$5$5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$17$17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.12.0.1$x^{12} + 3 x^{2} - 2 x + 5$$1$$12$$0$$C_{12}$$[\ ]^{12}$
$29$$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\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$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.3.2.1$x^{3} - 29$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
29.3.0.1$x^{3} - x + 3$$1$$3$$0$$C_3$$[\ ]^{3}$
29.3.0.1$x^{3} - x + 3$$1$$3$$0$$C_3$$[\ ]^{3}$
29.3.2.1$x^{3} - 29$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
721381Data not computed