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Properties

Label 20.12.1387878212...2688.1
Degree $20$
Signature $[12, 4]$
Discriminant $2^{20}\cdot 13^{10}\cdot 277^{9}$
Root discriminant $90.60$
Ramified primes $2, 13, 277$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 20T138

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![277, 0, 371, 0, -5697, 0, -5030, 0, 13437, 0, 7232, 0, -8928, 0, -613, 0, 583, 0, -50, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 50*x^18 + 583*x^16 - 613*x^14 - 8928*x^12 + 7232*x^10 + 13437*x^8 - 5030*x^6 - 5697*x^4 + 371*x^2 + 277)
 
gp: K = bnfinit(x^20 - 50*x^18 + 583*x^16 - 613*x^14 - 8928*x^12 + 7232*x^10 + 13437*x^8 - 5030*x^6 - 5697*x^4 + 371*x^2 + 277, 1)
 

Normalized defining polynomial

\( x^{20} - 50 x^{18} + 583 x^{16} - 613 x^{14} - 8928 x^{12} + 7232 x^{10} + 13437 x^{8} - 5030 x^{6} - 5697 x^{4} + 371 x^{2} + 277 \)

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:  \(1387878212018685329902859942836267122688=2^{20}\cdot 13^{10}\cdot 277^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $90.60$
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:  $2, 13, 277$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{13} a^{16} + \frac{6}{13} a^{14} + \frac{5}{13} a^{12} - \frac{6}{13} a^{10} - \frac{2}{13} a^{8} - \frac{6}{13} a^{6} + \frac{5}{13} a^{4} + \frac{6}{13} a^{2} + \frac{1}{13}$, $\frac{1}{13} a^{17} + \frac{6}{13} a^{15} + \frac{5}{13} a^{13} - \frac{6}{13} a^{11} - \frac{2}{13} a^{9} - \frac{6}{13} a^{7} + \frac{5}{13} a^{5} + \frac{6}{13} a^{3} + \frac{1}{13} a$, $\frac{1}{454790243321947443547} a^{18} + \frac{5657684475912272434}{454790243321947443547} a^{16} - \frac{28612920736942773129}{454790243321947443547} a^{14} + \frac{94096926222460740513}{454790243321947443547} a^{12} - \frac{67769054952236287918}{454790243321947443547} a^{10} + \frac{59404357327056502051}{454790243321947443547} a^{8} - \frac{52563176900676630631}{454790243321947443547} a^{6} + \frac{26299455756942944197}{454790243321947443547} a^{4} - \frac{153389946670308953767}{454790243321947443547} a^{2} + \frac{42818220333068610925}{454790243321947443547}$, $\frac{1}{454790243321947443547} a^{19} + \frac{5657684475912272434}{454790243321947443547} a^{17} - \frac{28612920736942773129}{454790243321947443547} a^{15} + \frac{94096926222460740513}{454790243321947443547} a^{13} - \frac{67769054952236287918}{454790243321947443547} a^{11} + \frac{59404357327056502051}{454790243321947443547} a^{9} - \frac{52563176900676630631}{454790243321947443547} a^{7} + \frac{26299455756942944197}{454790243321947443547} a^{5} - \frac{153389946670308953767}{454790243321947443547} a^{3} + \frac{42818220333068610925}{454790243321947443547} a$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 1445569259250 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T138:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 640
The 28 conjugacy class representatives for t20n138
Character table for t20n138 is not computed

Intermediate fields

\(\Q(\sqrt{13}) \), 5.5.12967201.1, 10.10.2185927923067213.1

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 R ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{6}$ $20$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{5}$ $20$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{6}$

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
2Data not computed
$13$13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
277Data not computed