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Properties

Label 20.12.3828204809...0625.4
Degree $20$
Signature $[12, 4]$
Discriminant $5^{10}\cdot 19^{6}\cdot 1699^{4}$
Root discriminant $23.94$
Ramified primes $5, 19, 1699$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T760

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -3, -15, 31, 47, 0, -36, -489, 278, 1116, -1070, -569, 1153, -343, -333, 318, -72, -38, 31, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 9*x^19 + 31*x^18 - 38*x^17 - 72*x^16 + 318*x^15 - 333*x^14 - 343*x^13 + 1153*x^12 - 569*x^11 - 1070*x^10 + 1116*x^9 + 278*x^8 - 489*x^7 - 36*x^6 + 47*x^4 + 31*x^3 - 15*x^2 - 3*x + 1)
 
gp: K = bnfinit(x^20 - 9*x^19 + 31*x^18 - 38*x^17 - 72*x^16 + 318*x^15 - 333*x^14 - 343*x^13 + 1153*x^12 - 569*x^11 - 1070*x^10 + 1116*x^9 + 278*x^8 - 489*x^7 - 36*x^6 + 47*x^4 + 31*x^3 - 15*x^2 - 3*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 9 x^{19} + 31 x^{18} - 38 x^{17} - 72 x^{16} + 318 x^{15} - 333 x^{14} - 343 x^{13} + 1153 x^{12} - 569 x^{11} - 1070 x^{10} + 1116 x^{9} + 278 x^{8} - 489 x^{7} - 36 x^{6} + 47 x^{4} + 31 x^{3} - 15 x^{2} - 3 x + 1 \)

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:  \(3828204809593746045712890625=5^{10}\cdot 19^{6}\cdot 1699^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.94$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{19} a^{16} + \frac{1}{19} a^{15} - \frac{2}{19} a^{14} + \frac{9}{19} a^{13} - \frac{9}{19} a^{12} + \frac{6}{19} a^{11} - \frac{8}{19} a^{10} + \frac{8}{19} a^{9} + \frac{7}{19} a^{8} - \frac{2}{19} a^{7} - \frac{2}{19} a^{5} - \frac{3}{19} a^{4} - \frac{6}{19} a^{3} + \frac{3}{19} a^{2} + \frac{5}{19} a - \frac{4}{19}$, $\frac{1}{19} a^{17} - \frac{3}{19} a^{15} - \frac{8}{19} a^{14} + \frac{1}{19} a^{13} - \frac{4}{19} a^{12} + \frac{5}{19} a^{11} - \frac{3}{19} a^{10} - \frac{1}{19} a^{9} - \frac{9}{19} a^{8} + \frac{2}{19} a^{7} - \frac{2}{19} a^{6} - \frac{1}{19} a^{5} - \frac{3}{19} a^{4} + \frac{9}{19} a^{3} + \frac{2}{19} a^{2} - \frac{9}{19} a + \frac{4}{19}$, $\frac{1}{19} a^{18} - \frac{5}{19} a^{15} - \frac{5}{19} a^{14} + \frac{4}{19} a^{13} - \frac{3}{19} a^{12} - \frac{4}{19} a^{11} - \frac{6}{19} a^{10} - \frac{4}{19} a^{9} + \frac{4}{19} a^{8} - \frac{8}{19} a^{7} - \frac{1}{19} a^{6} - \frac{9}{19} a^{5} + \frac{3}{19} a^{3} + \frac{7}{19}$, $\frac{1}{237481082406401} a^{19} - \frac{5217933553869}{237481082406401} a^{18} - \frac{476695287270}{237481082406401} a^{17} + \frac{3369049990725}{237481082406401} a^{16} - \frac{20953846800608}{237481082406401} a^{15} - \frac{51716478984014}{237481082406401} a^{14} - \frac{80224643898295}{237481082406401} a^{13} - \frac{104112058254770}{237481082406401} a^{12} - \frac{51783446507390}{237481082406401} a^{11} + \frac{114997654082376}{237481082406401} a^{10} + \frac{16890469176313}{237481082406401} a^{9} + \frac{114351738335915}{237481082406401} a^{8} + \frac{39875960686801}{237481082406401} a^{7} + \frac{86323601850175}{237481082406401} a^{6} + \frac{90393198426463}{237481082406401} a^{5} - \frac{74170560780998}{237481082406401} a^{4} + \frac{54456010509030}{237481082406401} a^{3} - \frac{57661881188178}{237481082406401} a^{2} - \frac{67450127050546}{237481082406401} a + \frac{97175228000538}{237481082406401}$

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:  \( 3812177.30689 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T760:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 102400
The 130 conjugacy class representatives for t20n760 are not computed
Character table for t20n760 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$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$19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$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}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
1699Data not computed