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Properties

Label 20.20.2858987213...6449.1
Degree $20$
Signature $[20, 0]$
Discriminant $19^{10}\cdot 293^{10}$
Root discriminant $74.61$
Ramified primes $19, 293$
Class number $6$ (GRH)
Class group $[6]$ (GRH)
Galois group $A_5$ (as 20T15)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -51, -177, 2805, -4435, -20117, 73078, -57223, -74283, 134055, -12858, -79221, 28417, 20641, -9033, -2859, 1122, 200, -57, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 - 57*x^18 + 200*x^17 + 1122*x^16 - 2859*x^15 - 9033*x^14 + 20641*x^13 + 28417*x^12 - 79221*x^11 - 12858*x^10 + 134055*x^9 - 74283*x^8 - 57223*x^7 + 73078*x^6 - 20117*x^5 - 4435*x^4 + 2805*x^3 - 177*x^2 - 51*x + 1)
 
gp: K = bnfinit(x^20 - 5*x^19 - 57*x^18 + 200*x^17 + 1122*x^16 - 2859*x^15 - 9033*x^14 + 20641*x^13 + 28417*x^12 - 79221*x^11 - 12858*x^10 + 134055*x^9 - 74283*x^8 - 57223*x^7 + 73078*x^6 - 20117*x^5 - 4435*x^4 + 2805*x^3 - 177*x^2 - 51*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} - 57 x^{18} + 200 x^{17} + 1122 x^{16} - 2859 x^{15} - 9033 x^{14} + 20641 x^{13} + 28417 x^{12} - 79221 x^{11} - 12858 x^{10} + 134055 x^{9} - 74283 x^{8} - 57223 x^{7} + 73078 x^{6} - 20117 x^{5} - 4435 x^{4} + 2805 x^{3} - 177 x^{2} - 51 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[20, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(28589872137663846814553020398335096449=19^{10}\cdot 293^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $74.61$
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:  $19, 293$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{14} - \frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{223171587548414896086665312458} a^{19} + \frac{8722545380710551643872005374}{111585793774207448043332656229} a^{18} - \frac{22493703020649573376196983642}{111585793774207448043332656229} a^{17} - \frac{37809830542225888814475999181}{223171587548414896086665312458} a^{16} - \frac{14632033490147522036887927137}{111585793774207448043332656229} a^{15} - \frac{96048083551351806958074481159}{223171587548414896086665312458} a^{14} - \frac{74890114081140991726325261971}{223171587548414896086665312458} a^{13} + \frac{13615733986518822819992506832}{111585793774207448043332656229} a^{12} - \frac{6873882176227050531020722165}{223171587548414896086665312458} a^{11} + \frac{14233027336077123913565524310}{111585793774207448043332656229} a^{10} + \frac{12881887484669592964493777202}{111585793774207448043332656229} a^{9} + \frac{64838593813142351332718150847}{223171587548414896086665312458} a^{8} + \frac{53377583198034321686290301749}{111585793774207448043332656229} a^{7} + \frac{52506037122111209513908479065}{223171587548414896086665312458} a^{6} + \frac{40225084074682734236724563691}{111585793774207448043332656229} a^{5} + \frac{24991082375024437954723405529}{111585793774207448043332656229} a^{4} + \frac{37902342806420035046181460971}{111585793774207448043332656229} a^{3} + \frac{15819870553762247724566635571}{223171587548414896086665312458} a^{2} - \frac{73631042326833217142279282689}{223171587548414896086665312458} a + \frac{42662724661823866638636291071}{111585793774207448043332656229}$

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

Class group and class number

$C_{6}$, which has order $6$ (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:  $19$
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:  \( 586413993994 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$A_5$ (as 20T15):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 60
The 5 conjugacy class representatives for $A_5$
Character table for $A_5$

Intermediate fields

5.5.30991489.2 x2, 10.10.960472390437121.1

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

Sibling fields

Degree 5 sibling: 5.5.30991489.2
Degree 6 sibling: 6.6.30991489.1
Degree 10 sibling: 10.10.960472390437121.1
Degree 12 sibling: Deg 12
Degree 15 sibling: 15.15.29766469523035740663169.1
Degree 30 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 ${\href{/LocalNumberField/2.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$19$19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
293Data not computed