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Properties

Label 20.0.56359149304...0625.1
Degree $20$
Signature $[0, 10]$
Discriminant $5^{10}\cdot 467^{2}\cdot 514417^{2}$
Root discriminant $15.40$
Ramified primes $5, 467, 514417$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1021

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 3, 0, -13, -10, 1, 11, 14, 56, -14, -43, -40, 125, -27, -28, 28, -10, 4, 2, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 2*x^18 + 4*x^17 - 10*x^16 + 28*x^15 - 28*x^14 - 27*x^13 + 125*x^12 - 40*x^11 - 43*x^10 - 14*x^9 + 56*x^8 + 14*x^7 + 11*x^6 + x^5 - 10*x^4 - 13*x^3 + 3*x + 1)
 
gp: K = bnfinit(x^20 - 2*x^19 + 2*x^18 + 4*x^17 - 10*x^16 + 28*x^15 - 28*x^14 - 27*x^13 + 125*x^12 - 40*x^11 - 43*x^10 - 14*x^9 + 56*x^8 + 14*x^7 + 11*x^6 + x^5 - 10*x^4 - 13*x^3 + 3*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} + 2 x^{18} + 4 x^{17} - 10 x^{16} + 28 x^{15} - 28 x^{14} - 27 x^{13} + 125 x^{12} - 40 x^{11} - 43 x^{10} - 14 x^{9} + 56 x^{8} + 14 x^{7} + 11 x^{6} + x^{5} - 10 x^{4} - 13 x^{3} + 3 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(563591493041426962890625=5^{10}\cdot 467^{2}\cdot 514417^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $15.40$
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:  $5, 467, 514417$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{3789985564396999291} a^{19} + \frac{456196154130323470}{3789985564396999291} a^{18} + \frac{12047025749224892}{3789985564396999291} a^{17} - \frac{499332945037205773}{3789985564396999291} a^{16} - \frac{877388999262920140}{3789985564396999291} a^{15} - \frac{1679120805513787776}{3789985564396999291} a^{14} + \frac{1636249583582439583}{3789985564396999291} a^{13} - \frac{1699641309840110529}{3789985564396999291} a^{12} - \frac{1070775466889230838}{3789985564396999291} a^{11} + \frac{5437302312556666}{56566948722343273} a^{10} + \frac{1480374964248530397}{3789985564396999291} a^{9} + \frac{1139850843120436889}{3789985564396999291} a^{8} + \frac{1399900133186502939}{3789985564396999291} a^{7} - \frac{1719413664966799178}{3789985564396999291} a^{6} + \frac{1363450093787865875}{3789985564396999291} a^{5} + \frac{812468987257367346}{3789985564396999291} a^{4} + \frac{1206062135165062383}{3789985564396999291} a^{3} - \frac{760657153962798932}{3789985564396999291} a^{2} + \frac{497853153524362939}{3789985564396999291} a + \frac{1540818909869961739}{3789985564396999291}$

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:  $9$
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:  \( 3018.34063127 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1021:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7257600
The 84 conjugacy class representatives for t20n1021 are not computed
Character table for t20n1021 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.0.240232739.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 20 sibling: data not computed
Degree 40 sibling: 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/7.14.0.1}{14} }{,}\,{\href{/LocalNumberField/7.6.0.1}{6} }$ ${\href{/LocalNumberField/11.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{10}$ ${\href{/LocalNumberField/43.14.0.1}{14} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}$ $18{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.14.0.1}{14} }{,}\,{\href{/LocalNumberField/53.6.0.1}{6} }$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{6}$

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$5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.14.7.1$x^{14} - 250 x^{8} + 15625 x^{2} - 312500$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
467Data not computed
514417Data not computed