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Properties

Label 20.10.1801695282...6875.2
Degree $20$
Signature $[10, 5]$
Discriminant $-\,5^{10}\cdot 19^{4}\cdot 1699^{5}$
Root discriminant $25.87$
Ramified primes $5, 19, 1699$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T876

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![19, -133, 312, -159, -564, 1316, -1272, -204, 2569, -3882, 3217, -1300, -393, 738, -349, 116, -27, -13, 12, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 12*x^18 - 13*x^17 - 27*x^16 + 116*x^15 - 349*x^14 + 738*x^13 - 393*x^12 - 1300*x^11 + 3217*x^10 - 3882*x^9 + 2569*x^8 - 204*x^7 - 1272*x^6 + 1316*x^5 - 564*x^4 - 159*x^3 + 312*x^2 - 133*x + 19)
 
gp: K = bnfinit(x^20 - 5*x^19 + 12*x^18 - 13*x^17 - 27*x^16 + 116*x^15 - 349*x^14 + 738*x^13 - 393*x^12 - 1300*x^11 + 3217*x^10 - 3882*x^9 + 2569*x^8 - 204*x^7 - 1272*x^6 + 1316*x^5 - 564*x^4 - 159*x^3 + 312*x^2 - 133*x + 19, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} + 12 x^{18} - 13 x^{17} - 27 x^{16} + 116 x^{15} - 349 x^{14} + 738 x^{13} - 393 x^{12} - 1300 x^{11} + 3217 x^{10} - 3882 x^{9} + 2569 x^{8} - 204 x^{7} - 1272 x^{6} + 1316 x^{5} - 564 x^{4} - 159 x^{3} + 312 x^{2} - 133 x + 19 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-18016952829639264630654296875=-\,5^{10}\cdot 19^{4}\cdot 1699^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $25.87$
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:  $5, 19, 1699$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{2751263205277450097673121} a^{19} - \frac{550164318602647551931964}{2751263205277450097673121} a^{18} - \frac{1053722296277404417039067}{2751263205277450097673121} a^{17} + \frac{932817140588838657032220}{2751263205277450097673121} a^{16} - \frac{487102621321977469753571}{2751263205277450097673121} a^{15} + \frac{988963719556282355264013}{2751263205277450097673121} a^{14} - \frac{153071235853028794518149}{2751263205277450097673121} a^{13} + \frac{206050598673246333629749}{2751263205277450097673121} a^{12} + \frac{824200968636838906260290}{2751263205277450097673121} a^{11} + \frac{415033756075192800350067}{2751263205277450097673121} a^{10} - \frac{904080749324740534414663}{2751263205277450097673121} a^{9} - \frac{375552269919643116158165}{2751263205277450097673121} a^{8} - \frac{815931156980260025099904}{2751263205277450097673121} a^{7} - \frac{162625482525251876800084}{2751263205277450097673121} a^{6} + \frac{543651736071115079674661}{2751263205277450097673121} a^{5} - \frac{566866783866752495332653}{2751263205277450097673121} a^{4} + \frac{843486589345708038681889}{2751263205277450097673121} a^{3} - \frac{924342130560051698886198}{2751263205277450097673121} a^{2} + \frac{1161624879411463905788112}{2751263205277450097673121} a + \frac{474443572038496358327780}{2751263205277450097673121}$

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:  $14$
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:  \( 5984182.47911 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T876:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 204800
The 230 conjugacy class representatives for t20n876 are not computed
Character table for t20n876 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.10.3256446753125.1

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 ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }$ $20$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}$

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}$
$19$$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.4.0.1$x^{4} - 2 x + 10$$1$$4$$0$$C_4$$[\ ]^{4}$
1699Data not computed