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Properties

Label 20.12.4340738075...0000.1
Degree $20$
Signature $[12, 4]$
Discriminant $2^{20}\cdot 3^{12}\cdot 5^{16}\cdot 691^{4}\cdot 5309^{6}$
Root discriminant $679.01$
Ramified primes $2, 3, 5, 691, 5309$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 20T1025

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![349861680657101806869390625, 0, -10873455887817253368515625, 0, 137377792648080355178125, 0, -933272710766328395625, 0, 3834786993819768625, 0, -10102154439305000, 0, 17469170847825, 0, -19736095850, 0, 14020930, 0, -5680, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5680*x^18 + 14020930*x^16 - 19736095850*x^14 + 17469170847825*x^12 - 10102154439305000*x^10 + 3834786993819768625*x^8 - 933272710766328395625*x^6 + 137377792648080355178125*x^4 - 10873455887817253368515625*x^2 + 349861680657101806869390625)
 
gp: K = bnfinit(x^20 - 5680*x^18 + 14020930*x^16 - 19736095850*x^14 + 17469170847825*x^12 - 10102154439305000*x^10 + 3834786993819768625*x^8 - 933272710766328395625*x^6 + 137377792648080355178125*x^4 - 10873455887817253368515625*x^2 + 349861680657101806869390625, 1)
 

Normalized defining polynomial

\( x^{20} - 5680 x^{18} + 14020930 x^{16} - 19736095850 x^{14} + 17469170847825 x^{12} - 10102154439305000 x^{10} + 3834786993819768625 x^{8} - 933272710766328395625 x^{6} + 137377792648080355178125 x^{4} - 10873455887817253368515625 x^{2} + 349861680657101806869390625 \)

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:  \(434073807522026850838993805941780627654560000000000000000=2^{20}\cdot 3^{12}\cdot 5^{16}\cdot 691^{4}\cdot 5309^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $679.01$
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:  $2, 3, 5, 691, 5309$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{5} a^{4}$, $\frac{1}{5} a^{5}$, $\frac{1}{5} a^{6}$, $\frac{1}{5} a^{7}$, $\frac{1}{25} a^{8}$, $\frac{1}{25} a^{9}$, $\frac{1}{132725} a^{10} - \frac{371}{132725} a^{8} + \frac{1034}{26545} a^{6} + \frac{1094}{26545} a^{4} + \frac{2067}{5309} a^{2}$, $\frac{1}{132725} a^{11} - \frac{371}{132725} a^{9} + \frac{1034}{26545} a^{7} + \frac{1094}{26545} a^{5} + \frac{2067}{5309} a^{3}$, $\frac{1}{3523185125} a^{12} - \frac{1136}{704637025} a^{10} + \frac{2804186}{704637025} a^{8} - \frac{250366}{140927405} a^{6} - \frac{7611039}{140927405} a^{4} + \frac{1664}{5309} a^{2}$, $\frac{1}{3523185125} a^{13} - \frac{1136}{704637025} a^{11} + \frac{2804186}{704637025} a^{9} - \frac{250366}{140927405} a^{7} - \frac{7611039}{140927405} a^{5} + \frac{1664}{5309} a^{3}$, $\frac{1}{56113769485875} a^{14} - \frac{1136}{11222753897175} a^{12} + \frac{2804186}{11222753897175} a^{10} + \frac{145689499459}{11222753897175} a^{8} - \frac{49416759232}{2244550779435} a^{6} - \frac{40812581}{422782215} a^{4} + \frac{889}{5309} a^{2} - \frac{1}{3}$, $\frac{1}{280568847429375} a^{15} - \frac{1136}{56113769485875} a^{13} + \frac{2804186}{56113769485875} a^{11} - \frac{150426162463}{11222753897175} a^{9} - \frac{49416759232}{11222753897175} a^{7} + \frac{25660061}{422782215} a^{5} + \frac{6198}{26545} a^{3} + \frac{1}{3} a$, $\frac{1}{1489540011002551875} a^{16} - \frac{1136}{297908002200510375} a^{14} + \frac{2804186}{297908002200510375} a^{12} - \frac{150426162463}{59581600440102075} a^{10} + \frac{850635328646633}{59581600440102075} a^{8} - \frac{44366472514}{2244550779435} a^{6} + \frac{13257462}{140927405} a^{4} - \frac{224}{15927} a^{2}$, $\frac{1}{1489540011002551875} a^{17} - \frac{371}{1489540011002551875} a^{15} - \frac{3226838}{297908002200510375} a^{13} - \frac{737243388841}{297908002200510375} a^{11} + \frac{17340944043522}{19860533480034025} a^{9} - \frac{90416373934}{3740917965725} a^{7} - \frac{19123996}{422782215} a^{5} + \frac{17474}{79635} a^{3} + \frac{1}{3} a$, $\frac{1}{6959646402229167233827899649227833797452984040803449259375} a^{18} - \frac{49070325145617491607623154402069585847}{198847040063690492395082847120795251355799544022955693125} a^{16} + \frac{1259472554702493662820392808299513172112463}{198847040063690492395082847120795251355799544022955693125} a^{14} - \frac{17995782346972929312518332755532069555596728551}{278385856089166689353115985969113351898119361632137970375} a^{12} - \frac{924608422250320503056552514553432586440149356472262}{278385856089166689353115985969113351898119361632137970375} a^{10} - \frac{195844915124900479457628924513833932222863594597446}{10487317991680794475536484685218058086197753310685175} a^{8} - \frac{89003932043328324361218312821772089733748914791}{1975384816666188448961477620120184231719298043075} a^{6} - \frac{109939881022962499064549723908604914630714}{4961097046942144657403095674314520579441949} a^{4} - \frac{670825456433203027498663701928173761914}{2803407636245325668150176497069798782883} a^{2} + \frac{54557982478091650019698907489116951}{528048151487158724458499999448069087}$, $\frac{1}{6959646402229167233827899649227833797452984040803449259375} a^{19} - \frac{49070325145617491607623154402069585847}{198847040063690492395082847120795251355799544022955693125} a^{17} - \frac{52661248141277794050640080501545188290317}{66282346687896830798360949040265083785266514674318564375} a^{15} - \frac{2241389952106792216966166461163159599701540141}{92795285363055563117705328656371117299373120544045990125} a^{13} - \frac{190486420003534702554616325809315852570263071773858}{55677171217833337870623197193822670379623872326427594075} a^{11} + \frac{17058463980333792946007878354612652910717657283288}{2097463598336158895107296937043611617239550662137035} a^{9} - \frac{4773843097779951160688602008180478739601880489}{131692321111079229930765174674678948781286536205} a^{7} + \frac{4201027740083507428716173194756208615775599}{74416455704132169861046435114717808691629235} a^{5} + \frac{4117226013225793654869997982549860702393}{14017038181226628340750882485348993914415} a^{3} + \frac{230574032973811224839198907305139980}{528048151487158724458499999448069087} a$

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:  \( 1045386115320000000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1025:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7372800
The 216 conjugacy class representatives for t20n1025 are not computed
Character table for t20n1025 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.6.5438807015625.1

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

Sibling fields

Degree 20 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 R R R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ $16{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ $16{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}$ $16{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }$ $16{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
2Data not computed
$3$3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.6.6.5$x^{6} + 6 x^{3} + 9 x^{2} + 9$$3$$2$$6$$S_3^2$$[3/2, 3/2]_{2}^{2}$
3.6.6.5$x^{6} + 6 x^{3} + 9 x^{2} + 9$$3$$2$$6$$S_3^2$$[3/2, 3/2]_{2}^{2}$
$5$5.8.7.1$x^{8} - 5$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
5.12.9.1$x^{12} - 10 x^{8} - 375 x^{4} - 2000$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
691Data not computed
5309Data not computed