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Properties

Label 20.4.57207170613...5625.1
Degree $20$
Signature $[4, 8]$
Discriminant $5^{10}\cdot 53^{8}\cdot 97^{2}$
Root discriminant $17.29$
Ramified primes $5, 53, 97$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T277

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -5, -8, 6, 20, -31, -35, 73, -14, -58, 103, -71, 20, 33, -87, 85, -39, 7, 4, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 4*x^18 + 7*x^17 - 39*x^16 + 85*x^15 - 87*x^14 + 33*x^13 + 20*x^12 - 71*x^11 + 103*x^10 - 58*x^9 - 14*x^8 + 73*x^7 - 35*x^6 - 31*x^5 + 20*x^4 + 6*x^3 - 8*x^2 - 5*x - 1)
 
gp: K = bnfinit(x^20 - 4*x^19 + 4*x^18 + 7*x^17 - 39*x^16 + 85*x^15 - 87*x^14 + 33*x^13 + 20*x^12 - 71*x^11 + 103*x^10 - 58*x^9 - 14*x^8 + 73*x^7 - 35*x^6 - 31*x^5 + 20*x^4 + 6*x^3 - 8*x^2 - 5*x - 1, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 4 x^{18} + 7 x^{17} - 39 x^{16} + 85 x^{15} - 87 x^{14} + 33 x^{13} + 20 x^{12} - 71 x^{11} + 103 x^{10} - 58 x^{9} - 14 x^{8} + 73 x^{7} - 35 x^{6} - 31 x^{5} + 20 x^{4} + 6 x^{3} - 8 x^{2} - 5 x - 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(5720717061332965322265625=5^{10}\cdot 53^{8}\cdot 97^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $17.29$
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:  $5, 53, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{4} a^{16} - \frac{1}{4} a^{14} - \frac{1}{4} a^{13} + \frac{1}{4} a^{12} - \frac{1}{2} a^{11} + \frac{1}{4} a^{10} - \frac{1}{2} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{4} a^{17} - \frac{1}{4} a^{15} - \frac{1}{4} a^{14} + \frac{1}{4} a^{13} - \frac{1}{2} a^{12} + \frac{1}{4} a^{11} - \frac{1}{2} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{6} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{4} a$, $\frac{1}{16} a^{18} - \frac{1}{16} a^{17} - \frac{1}{8} a^{16} - \frac{1}{4} a^{15} + \frac{3}{16} a^{14} + \frac{1}{8} a^{13} - \frac{1}{8} a^{12} - \frac{3}{16} a^{11} + \frac{1}{16} a^{10} - \frac{3}{16} a^{9} - \frac{5}{16} a^{8} - \frac{7}{16} a^{6} + \frac{1}{4} a^{5} + \frac{3}{8} a^{4} + \frac{7}{16} a^{3} - \frac{1}{16} a^{2} + \frac{3}{8} a + \frac{5}{16}$, $\frac{1}{31352627259664} a^{19} + \frac{236442647393}{7838156814916} a^{18} - \frac{404763501215}{31352627259664} a^{17} + \frac{1468006116941}{15676313629832} a^{16} - \frac{557982433345}{31352627259664} a^{15} - \frac{6433726204831}{31352627259664} a^{14} - \frac{392774343083}{3919078407458} a^{13} + \frac{5309403624763}{31352627259664} a^{12} + \frac{5473195104085}{15676313629832} a^{11} - \frac{3265803460287}{15676313629832} a^{10} + \frac{373704013227}{7838156814916} a^{9} + \frac{10546933513791}{31352627259664} a^{8} - \frac{13460420554319}{31352627259664} a^{7} + \frac{14999634367233}{31352627259664} a^{6} + \frac{2376831078017}{15676313629832} a^{5} + \frac{927108072381}{31352627259664} a^{4} + \frac{6020619360229}{15676313629832} a^{3} - \frac{4225559396191}{31352627259664} a^{2} - \frac{10371572931477}{31352627259664} a + \frac{6413119315465}{31352627259664}$

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:  $11$
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:  \( 53732.3155396 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T277:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 3840
The 48 conjugacy class representatives for t20n277
Character table for t20n277 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.1.70225.1, 10.2.478360410625.1, 10.2.24657753125.1, 10.2.2391802053125.1

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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$53$53.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
53.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
53.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
53.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
53.6.4.1$x^{6} + 742 x^{3} + 351125$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
53.6.4.1$x^{6} + 742 x^{3} + 351125$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
$97$97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.4.2.1$x^{4} + 873 x^{2} + 235225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
97.6.0.1$x^{6} - x + 10$$1$$6$$0$$C_6$$[\ ]^{6}$
97.6.0.1$x^{6} - x + 10$$1$$6$$0$$C_6$$[\ ]^{6}$