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Properties

Label 20.12.1947870881...3125.1
Degree $20$
Signature $[12, 4]$
Discriminant $5^{10}\cdot 11^{8}\cdot 13^{8}\cdot 409\cdot 2789$
Root discriminant $32.69$
Ramified primes $5, 11, 13, 409, 2789$
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, -6, 13, 91, -34, -417, 19, 905, 62, -1228, -57, 1141, -110, -687, 203, 232, -131, -21, 34, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 34*x^18 - 21*x^17 - 131*x^16 + 232*x^15 + 203*x^14 - 687*x^13 - 110*x^12 + 1141*x^11 - 57*x^10 - 1228*x^9 + 62*x^8 + 905*x^7 + 19*x^6 - 417*x^5 - 34*x^4 + 91*x^3 + 13*x^2 - 6*x - 1)
 
gp: K = bnfinit(x^20 - 10*x^19 + 34*x^18 - 21*x^17 - 131*x^16 + 232*x^15 + 203*x^14 - 687*x^13 - 110*x^12 + 1141*x^11 - 57*x^10 - 1228*x^9 + 62*x^8 + 905*x^7 + 19*x^6 - 417*x^5 - 34*x^4 + 91*x^3 + 13*x^2 - 6*x - 1, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 34 x^{18} - 21 x^{17} - 131 x^{16} + 232 x^{15} + 203 x^{14} - 687 x^{13} - 110 x^{12} + 1141 x^{11} - 57 x^{10} - 1228 x^{9} + 62 x^{8} + 905 x^{7} + 19 x^{6} - 417 x^{5} - 34 x^{4} + 91 x^{3} + 13 x^{2} - 6 x - 1 \)

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:  \(1947870881194502131483408203125=5^{10}\cdot 11^{8}\cdot 13^{8}\cdot 409\cdot 2789\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $32.69$
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:  $5, 11, 13, 409, 2789$
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 \(a\))

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

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:  \( 72632061.1392 \) (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

\(\Q(\sqrt{5}) \), 10.6.1306755003125.1

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 $20$ $20$ R ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ R R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$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}$
$11$11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.6.0.1$x^{6} + x^{2} - 2 x + 8$$1$$6$$0$$C_6$$[\ ]^{6}$
11.8.6.2$x^{8} - 781 x^{4} + 290521$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
$13$13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.6.4.3$x^{6} + 65 x^{3} + 1352$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
13.6.4.3$x^{6} + 65 x^{3} + 1352$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
409Data not computed
2789Data not computed