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Properties

Label 20.16.4017846883...0625.1
Degree $20$
Signature $[16, 2]$
Discriminant $5^{10}\cdot 419^{2}\cdot 695771^{4}$
Root discriminant $60.28$
Ramified primes $5, 419, 695771$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1030

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![302989, -1608972, 2216019, 574525, -2636562, 1541404, -996698, 476564, 641383, -498941, -77732, 69919, 14743, 6905, -6101, -2177, 926, 163, -53, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 - 53*x^18 + 163*x^17 + 926*x^16 - 2177*x^15 - 6101*x^14 + 6905*x^13 + 14743*x^12 + 69919*x^11 - 77732*x^10 - 498941*x^9 + 641383*x^8 + 476564*x^7 - 996698*x^6 + 1541404*x^5 - 2636562*x^4 + 574525*x^3 + 2216019*x^2 - 1608972*x + 302989)
 
gp: K = bnfinit(x^20 - 4*x^19 - 53*x^18 + 163*x^17 + 926*x^16 - 2177*x^15 - 6101*x^14 + 6905*x^13 + 14743*x^12 + 69919*x^11 - 77732*x^10 - 498941*x^9 + 641383*x^8 + 476564*x^7 - 996698*x^6 + 1541404*x^5 - 2636562*x^4 + 574525*x^3 + 2216019*x^2 - 1608972*x + 302989, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} - 53 x^{18} + 163 x^{17} + 926 x^{16} - 2177 x^{15} - 6101 x^{14} + 6905 x^{13} + 14743 x^{12} + 69919 x^{11} - 77732 x^{10} - 498941 x^{9} + 641383 x^{8} + 476564 x^{7} - 996698 x^{6} + 1541404 x^{5} - 2636562 x^{4} + 574525 x^{3} + 2216019 x^{2} - 1608972 x + 302989 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(401784688398260722928486785556640625=5^{10}\cdot 419^{2}\cdot 695771^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $60.28$
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:  $5, 419, 695771$
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}{77197131299197252112828454509367975086318707258697000393} a^{19} + \frac{15502727440482611674148399394343663673076234321400617281}{77197131299197252112828454509367975086318707258697000393} a^{18} - \frac{17170480827188620857339750721585808176802426495667242826}{77197131299197252112828454509367975086318707258697000393} a^{17} - \frac{23879203343794874638137533108963560348593416839306722154}{77197131299197252112828454509367975086318707258697000393} a^{16} + \frac{36792471580590961896636785152903382619630146546332683545}{77197131299197252112828454509367975086318707258697000393} a^{15} + \frac{21579932275558861765525488895292948185217072362887744149}{77197131299197252112828454509367975086318707258697000393} a^{14} - \frac{34320902439956052804508088314116187996023864428725084419}{77197131299197252112828454509367975086318707258697000393} a^{13} - \frac{7152843928125792882190417232317627791296410585518188540}{77197131299197252112828454509367975086318707258697000393} a^{12} + \frac{32187134059906771941981816700556780364460146185357671696}{77197131299197252112828454509367975086318707258697000393} a^{11} + \frac{14658139339050344660809646252631516098111667618453647081}{77197131299197252112828454509367975086318707258697000393} a^{10} - \frac{12222643742100484193333676447321787346210992030169547201}{77197131299197252112828454509367975086318707258697000393} a^{9} + \frac{28160923401095272521418318055067224942684672375504946931}{77197131299197252112828454509367975086318707258697000393} a^{8} - \frac{7904372004295695273364214109734612249210010951649656701}{77197131299197252112828454509367975086318707258697000393} a^{7} + \frac{27662894855488443822407952372203948722357605410383491965}{77197131299197252112828454509367975086318707258697000393} a^{6} + \frac{3714070470617545434654363729205714963094986893258935651}{77197131299197252112828454509367975086318707258697000393} a^{5} - \frac{23127287904674485185698684586729785139848338897083941077}{77197131299197252112828454509367975086318707258697000393} a^{4} - \frac{35237584569078950666224637351909953860170656466029475839}{77197131299197252112828454509367975086318707258697000393} a^{3} + \frac{31042709762614194859965798178113965557496592442887172630}{77197131299197252112828454509367975086318707258697000393} a^{2} + \frac{20507815386389524725625596852233133980836037101764311879}{77197131299197252112828454509367975086318707258697000393} a + \frac{12793350497078756829583531111018493306298790453088249283}{77197131299197252112828454509367975086318707258697000393}$

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:  $17$
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:  \( 88764287740.3 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1030:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7372800
The 189 conjugacy class representatives for t20n1030 are not computed
Character table for t20n1030 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.10.911025153125.1

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 32 sibling: 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.8.0.1}{8} }$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}$

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$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$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}$
419Data not computed
695771Data not computed