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Properties

Label 20.8.14735319462...0625.1
Degree $20$
Signature $[8, 6]$
Discriminant $5^{10}\cdot 1039^{2}\cdot 1049^{7}$
Root discriminant $51.10$
Ramified primes $5, 1039, 1049$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T781

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![860975209, 587290123, -1272386337, -290470403, 865682349, 76009560, -328689387, -15382586, 77335700, 2677732, -11886078, -370164, 1217625, 35541, -82643, -2132, 3581, 71, -90, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 - 90*x^18 + 71*x^17 + 3581*x^16 - 2132*x^15 - 82643*x^14 + 35541*x^13 + 1217625*x^12 - 370164*x^11 - 11886078*x^10 + 2677732*x^9 + 77335700*x^8 - 15382586*x^7 - 328689387*x^6 + 76009560*x^5 + 865682349*x^4 - 290470403*x^3 - 1272386337*x^2 + 587290123*x + 860975209)
 
gp: K = bnfinit(x^20 - x^19 - 90*x^18 + 71*x^17 + 3581*x^16 - 2132*x^15 - 82643*x^14 + 35541*x^13 + 1217625*x^12 - 370164*x^11 - 11886078*x^10 + 2677732*x^9 + 77335700*x^8 - 15382586*x^7 - 328689387*x^6 + 76009560*x^5 + 865682349*x^4 - 290470403*x^3 - 1272386337*x^2 + 587290123*x + 860975209, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} - 90 x^{18} + 71 x^{17} + 3581 x^{16} - 2132 x^{15} - 82643 x^{14} + 35541 x^{13} + 1217625 x^{12} - 370164 x^{11} - 11886078 x^{10} + 2677732 x^{9} + 77335700 x^{8} - 15382586 x^{7} - 328689387 x^{6} + 76009560 x^{5} + 865682349 x^{4} - 290470403 x^{3} - 1272386337 x^{2} + 587290123 x + 860975209 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(14735319462983715612393284462890625=5^{10}\cdot 1039^{2}\cdot 1049^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $51.10$
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:  $5, 1039, 1049$
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}{157503721436633036291585577098733485697999218551676806782756699425252609881} a^{19} - \frac{22263700050229711804466219043341897650479570651415542481668200000052734604}{157503721436633036291585577098733485697999218551676806782756699425252609881} a^{18} - \frac{44323115944545333935234276392252167709033455516654894824402350125787327417}{157503721436633036291585577098733485697999218551676806782756699425252609881} a^{17} - \frac{57193946769891314450495933212096902100265032081687214392016320283948521538}{157503721436633036291585577098733485697999218551676806782756699425252609881} a^{16} - \frac{67895752905299738919547435363159092934962673948052182616347142130865507787}{157503721436633036291585577098733485697999218551676806782756699425252609881} a^{15} - \frac{3752386394007404318487038279839663993318205332061937149920203118244885102}{157503721436633036291585577098733485697999218551676806782756699425252609881} a^{14} - \frac{9595925167909314652849975286761629268228207324985364896775608777556047401}{157503721436633036291585577098733485697999218551676806782756699425252609881} a^{13} - \frac{33958144834046161894157199672526271861916677959383995772666931609059759543}{157503721436633036291585577098733485697999218551676806782756699425252609881} a^{12} - \frac{34483773209119596216519775919351171806315395989742317205375998029077481043}{157503721436633036291585577098733485697999218551676806782756699425252609881} a^{11} - \frac{11532703226573215569614010469666615278062994959027858759848866393880333295}{157503721436633036291585577098733485697999218551676806782756699425252609881} a^{10} + \frac{66509753527174769914023154686724102288880046332250782047705138564046497587}{157503721436633036291585577098733485697999218551676806782756699425252609881} a^{9} + \frac{60454315946979614477840298214754663227961875337499862302721028887175696248}{157503721436633036291585577098733485697999218551676806782756699425252609881} a^{8} - \frac{32489114922574680990000375571681866634706863622321160689897557318149690104}{157503721436633036291585577098733485697999218551676806782756699425252609881} a^{7} - \frac{38250440851486134533544174567463545867895296076317003903296285132279092123}{157503721436633036291585577098733485697999218551676806782756699425252609881} a^{6} + \frac{47073545647286793001318042903150458099956354948354874206893297884914876888}{157503721436633036291585577098733485697999218551676806782756699425252609881} a^{5} + \frac{45737927119436367362023740071047207517346261257758457741046011261557543923}{157503721436633036291585577098733485697999218551676806782756699425252609881} a^{4} - \frac{77738008074841351989020655092491110448468493927451129463793016348288517522}{157503721436633036291585577098733485697999218551676806782756699425252609881} a^{3} + \frac{68929753364162093269146701194520592645872858559347141036170802949680554081}{157503721436633036291585577098733485697999218551676806782756699425252609881} a^{2} - \frac{74305141848647513539339695853554923621446095373823001044869997954266175903}{157503721436633036291585577098733485697999218551676806782756699425252609881} a + \frac{9202887831429623187814240779216085352142978255155019586522124811041961501}{157503721436633036291585577098733485697999218551676806782756699425252609881}$

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:  $13$
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:  \( 2605723972.29 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T781:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 115200
The 119 conjugacy class representatives for t20n781 are not computed
Character table for t20n781 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.26225.1, 10.4.3405971875.1

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 24 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ $20$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ $20$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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
$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}$
5.12.6.1$x^{12} + 500 x^{6} - 3125 x^{2} + 62500$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
1039Data not computed
1049Data not computed