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Properties

Label 20.4.30136170642...0000.1
Degree $20$
Signature $[4, 8]$
Discriminant $2^{20}\cdot 5^{11}\cdot 19^{4}\cdot 461^{4}$
Root discriminant $29.78$
Ramified primes $2, 5, 19, 461$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1010

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-5, 50, -85, -300, -1399, -2006, -2508, -602, 795, 1856, 937, 112, -245, -326, -56, 62, 65, -2, -15, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 15*x^18 - 2*x^17 + 65*x^16 + 62*x^15 - 56*x^14 - 326*x^13 - 245*x^12 + 112*x^11 + 937*x^10 + 1856*x^9 + 795*x^8 - 602*x^7 - 2508*x^6 - 2006*x^5 - 1399*x^4 - 300*x^3 - 85*x^2 + 50*x - 5)
 
gp: K = bnfinit(x^20 - 15*x^18 - 2*x^17 + 65*x^16 + 62*x^15 - 56*x^14 - 326*x^13 - 245*x^12 + 112*x^11 + 937*x^10 + 1856*x^9 + 795*x^8 - 602*x^7 - 2508*x^6 - 2006*x^5 - 1399*x^4 - 300*x^3 - 85*x^2 + 50*x - 5, 1)
 

Normalized defining polynomial

\( x^{20} - 15 x^{18} - 2 x^{17} + 65 x^{16} + 62 x^{15} - 56 x^{14} - 326 x^{13} - 245 x^{12} + 112 x^{11} + 937 x^{10} + 1856 x^{9} + 795 x^{8} - 602 x^{7} - 2508 x^{6} - 2006 x^{5} - 1399 x^{4} - 300 x^{3} - 85 x^{2} + 50 x - 5 \)

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:  \(301361706426703923200000000000=2^{20}\cdot 5^{11}\cdot 19^{4}\cdot 461^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $29.78$
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:  $2, 5, 19, 461$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{3} a^{17} + \frac{1}{3} a^{16} + \frac{1}{3} a^{15} + \frac{1}{3} a^{11} - \frac{1}{3} a^{10} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{18} - \frac{1}{3} a^{15} + \frac{1}{3} a^{12} + \frac{1}{3} a^{11} + \frac{1}{3} a^{10} - \frac{1}{3} a^{9} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{815587897120959020391333412023} a^{19} + \frac{74302990886910515221026376268}{815587897120959020391333412023} a^{18} - \frac{135332998794438086729263124161}{815587897120959020391333412023} a^{17} - \frac{194398771903258901815867721231}{815587897120959020391333412023} a^{16} - \frac{17889478538385341808114473910}{271862632373653006797111137341} a^{15} - \frac{133531408584315777519468851872}{271862632373653006797111137341} a^{14} - \frac{218285270351870993482550529083}{815587897120959020391333412023} a^{13} - \frac{6525650837312225306812606807}{271862632373653006797111137341} a^{12} - \frac{261163164204739349062059520375}{815587897120959020391333412023} a^{11} + \frac{179181943195300077186080024162}{815587897120959020391333412023} a^{10} - \frac{347415688023970860193903029680}{815587897120959020391333412023} a^{9} - \frac{107010911830691127113703570981}{271862632373653006797111137341} a^{8} + \frac{190422911181372096918199511950}{815587897120959020391333412023} a^{7} - \frac{75010906670351500161510741094}{271862632373653006797111137341} a^{6} + \frac{187215469931475896864818224700}{815587897120959020391333412023} a^{5} - \frac{27464778633329138042342493313}{271862632373653006797111137341} a^{4} + \frac{9846631867013056481489776007}{271862632373653006797111137341} a^{3} + \frac{120215713420340802370004908004}{271862632373653006797111137341} a^{2} + \frac{295588355928404950353703916984}{815587897120959020391333412023} a - \frac{37634802309850981076056096623}{271862632373653006797111137341}$

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:  \( 15115694.8774 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1010:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 3686400
The 180 conjugacy class representatives for t20n1010 are not computed
Character table for t20n1010 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.6.61376064800000.1

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

Sibling fields

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 ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ R $20$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ $20$ R ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
2Data not computed
$5$5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$19$19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$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$$[\ ]_{2}$
19.10.0.1$x^{10} + x^{2} - 2 x + 14$$1$$10$$0$$C_{10}$$[\ ]^{10}$
461Data not computed