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Properties

Label 20.0.12902360962...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{28}\cdot 5^{22}\cdot 17^{10}$
Root discriminant $63.90$
Ramified primes $2, 5, 17$
Class number Not computed
Class group Not computed
Galois group $C_2\times F_5$ (as 20T13)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![13948557461, 63672480, 693760140, 999109080, 7225, -383528, 25490700, -680, -30590, -340200, -170832, 720, 8270, 2080, 0, -8, 245, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 245*x^16 - 8*x^15 + 2080*x^13 + 8270*x^12 + 720*x^11 - 170832*x^10 - 340200*x^9 - 30590*x^8 - 680*x^7 + 25490700*x^6 - 383528*x^5 + 7225*x^4 + 999109080*x^3 + 693760140*x^2 + 63672480*x + 13948557461)
 
gp: K = bnfinit(x^20 + 245*x^16 - 8*x^15 + 2080*x^13 + 8270*x^12 + 720*x^11 - 170832*x^10 - 340200*x^9 - 30590*x^8 - 680*x^7 + 25490700*x^6 - 383528*x^5 + 7225*x^4 + 999109080*x^3 + 693760140*x^2 + 63672480*x + 13948557461, 1)
 

Normalized defining polynomial

\( x^{20} + 245 x^{16} - 8 x^{15} + 2080 x^{13} + 8270 x^{12} + 720 x^{11} - 170832 x^{10} - 340200 x^{9} - 30590 x^{8} - 680 x^{7} + 25490700 x^{6} - 383528 x^{5} + 7225 x^{4} + 999109080 x^{3} + 693760140 x^{2} + 63672480 x + 13948557461 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1290236096287360000000000000000000000=2^{28}\cdot 5^{22}\cdot 17^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $63.90$
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:  $2, 5, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{12} - \frac{1}{4} a^{10} - \frac{1}{2} a^{9} + \frac{1}{4} a^{8} + \frac{1}{4} a^{6} + \frac{1}{4} a^{4} - \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{4} a^{15} - \frac{1}{4} a^{13} - \frac{1}{4} a^{11} + \frac{1}{4} a^{9} - \frac{1}{2} a^{8} + \frac{1}{4} a^{7} + \frac{1}{4} a^{5} - \frac{1}{4} a^{3} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{12} a^{16} - \frac{1}{12} a^{15} + \frac{1}{12} a^{14} - \frac{1}{4} a^{13} - \frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{12} a^{10} + \frac{5}{12} a^{9} - \frac{1}{4} a^{8} - \frac{1}{12} a^{7} - \frac{5}{12} a^{6} + \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} + \frac{5}{12} a^{2} - \frac{1}{12} a - \frac{1}{3}$, $\frac{1}{12} a^{17} + \frac{1}{12} a^{14} - \frac{1}{4} a^{12} + \frac{1}{6} a^{11} + \frac{1}{12} a^{10} - \frac{1}{3} a^{9} + \frac{5}{12} a^{8} - \frac{1}{2} a^{7} + \frac{1}{12} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} - \frac{1}{3} a^{3} + \frac{1}{12} a^{2} + \frac{1}{12} a - \frac{1}{12}$, $\frac{1}{12} a^{18} + \frac{1}{12} a^{15} - \frac{1}{4} a^{13} + \frac{1}{6} a^{12} + \frac{1}{12} a^{11} + \frac{1}{6} a^{10} + \frac{5}{12} a^{9} + \frac{1}{12} a^{7} - \frac{1}{2} a^{6} + \frac{1}{4} a^{5} - \frac{1}{3} a^{4} + \frac{1}{12} a^{3} - \frac{5}{12} a^{2} - \frac{1}{12} a - \frac{1}{2}$, $\frac{1}{45071857900243400502669735791509542192480970128700232331386500184235449294943076870631604} a^{19} - \frac{958235213302119380429807559496510018477612007665580557517107079023171345037945631045185}{45071857900243400502669735791509542192480970128700232331386500184235449294943076870631604} a^{18} - \frac{126676556177333451286463577702760864096025436241935905334516365034248261953865453978435}{15023952633414466834223245263836514064160323376233410777128833394745149764981025623543868} a^{17} - \frac{57364129079123847347236822671203506643666723450592732790021223540576491338240248160830}{3755988158353616708555811315959128516040080844058352694282208348686287441245256405885967} a^{16} - \frac{563099781380167789819771962791974461320350122916240197679938236677913738960723177492560}{11267964475060850125667433947877385548120242532175058082846625046058862323735769217657901} a^{15} - \frac{1511379360679976028366305862722133098470947933886819697714075799114434555478151472073043}{45071857900243400502669735791509542192480970128700232331386500184235449294943076870631604} a^{14} - \frac{2101350559486703522571161377319903454333363993198057002950266185793442518991261254708569}{22535928950121700251334867895754771096240485064350116165693250092117724647471538435315802} a^{13} + \frac{1758111866729885157442462231166942091949749902649060688778876798461281045986438906581599}{15023952633414466834223245263836514064160323376233410777128833394745149764981025623543868} a^{12} - \frac{31473295998749201590808027880324492894482566702546276071386523273561418092684055141208}{3755988158353616708555811315959128516040080844058352694282208348686287441245256405885967} a^{11} - \frac{3707971906956616237894634515533428739976495613475520323904123563538539852414101239289353}{45071857900243400502669735791509542192480970128700232331386500184235449294943076870631604} a^{10} - \frac{3031843800660731961813634404045148398632910532907990073191428708071359356391095549748739}{7511976316707233417111622631918257032080161688116705388564416697372574882490512811771934} a^{9} - \frac{19927509061223726055461798314427449050794585876809234025324191699560495484770709438707719}{45071857900243400502669735791509542192480970128700232331386500184235449294943076870631604} a^{8} - \frac{3387554810088928261055888408374070225294786359830701347842802455270829938492490732530049}{11267964475060850125667433947877385548120242532175058082846625046058862323735769217657901} a^{7} - \frac{3859049273505123969045507045304619479047694006748954654748093785112289715367550439107693}{45071857900243400502669735791509542192480970128700232331386500184235449294943076870631604} a^{6} + \frac{7850848954235933555194792009675531359322707537352169918920467830971067884908532594159627}{22535928950121700251334867895754771096240485064350116165693250092117724647471538435315802} a^{5} - \frac{6399775036154262165678247071469852068174738527526266852660048450389592228264227457448869}{15023952633414466834223245263836514064160323376233410777128833394745149764981025623543868} a^{4} + \frac{9586539840946129957609320138386221090700768690830731337042953642803716868265411063732983}{45071857900243400502669735791509542192480970128700232331386500184235449294943076870631604} a^{3} - \frac{6645329489727797413761978464831161232994619584071972995093299851498966168500540917061321}{22535928950121700251334867895754771096240485064350116165693250092117724647471538435315802} a^{2} - \frac{3792489569949026861069750303820566132446057413394056092233478617292129691866486740335107}{15023952633414466834223245263836514064160323376233410777128833394745149764981025623543868} a + \frac{16336090501507292830684658949417760750195658463339860386387153694226041829918796388018485}{45071857900243400502669735791509542192480970128700232331386500184235449294943076870631604}$

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

Class group and class number

Not computed

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $9$
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:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times F_5$ (as 20T13):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 40
The 10 conjugacy class representatives for $C_2\times F_5$
Character table for $C_2\times F_5$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{-17}) \), \(\Q(\sqrt{-85}) \), \(\Q(\sqrt{5}, \sqrt{-17})\), 5.1.50000.1, 10.2.12500000000.1, 10.0.227177120000000000.1, 10.0.1135885600000000000.1

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

Sibling fields

Galois closure: data not computed
Degree 10 siblings: data not computed
Degree 20 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 R ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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
5Data not computed
$17$17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.8.4.1$x^{8} + 6358 x^{4} - 4913 x^{2} + 10106041$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
17.8.4.1$x^{8} + 6358 x^{4} - 4913 x^{2} + 10106041$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$