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Properties

Label 20.12.6634545047...0000.2
Degree $20$
Signature $[12, 4]$
Discriminant $2^{20}\cdot 5^{10}\cdot 79^{5}\cdot 4588681^{2}$
Root discriminant $61.81$
Ramified primes $2, 5, 79, 4588681$
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![1098795961, -1655050866, -1769937551, 1756330666, 1355308903, -722933074, -542617125, 156814362, 126327623, -20094922, -18422931, 1577342, 1749120, -74370, -108829, 1948, 4312, -22, -99, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 99*x^18 - 22*x^17 + 4312*x^16 + 1948*x^15 - 108829*x^14 - 74370*x^13 + 1749120*x^12 + 1577342*x^11 - 18422931*x^10 - 20094922*x^9 + 126327623*x^8 + 156814362*x^7 - 542617125*x^6 - 722933074*x^5 + 1355308903*x^4 + 1756330666*x^3 - 1769937551*x^2 - 1655050866*x + 1098795961)
 
gp: K = bnfinit(x^20 - 99*x^18 - 22*x^17 + 4312*x^16 + 1948*x^15 - 108829*x^14 - 74370*x^13 + 1749120*x^12 + 1577342*x^11 - 18422931*x^10 - 20094922*x^9 + 126327623*x^8 + 156814362*x^7 - 542617125*x^6 - 722933074*x^5 + 1355308903*x^4 + 1756330666*x^3 - 1769937551*x^2 - 1655050866*x + 1098795961, 1)
 

Normalized defining polynomial

\( x^{20} - 99 x^{18} - 22 x^{17} + 4312 x^{16} + 1948 x^{15} - 108829 x^{14} - 74370 x^{13} + 1749120 x^{12} + 1577342 x^{11} - 18422931 x^{10} - 20094922 x^{9} + 126327623 x^{8} + 156814362 x^{7} - 542617125 x^{6} - 722933074 x^{5} + 1355308903 x^{4} + 1756330666 x^{3} - 1769937551 x^{2} - 1655050866 x + 1098795961 \)

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:  \(663454504774367623174543360000000000=2^{20}\cdot 5^{10}\cdot 79^{5}\cdot 4588681^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $61.81$
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:  $2, 5, 79, 4588681$
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}{3021954301553833138809874076771984027243457302815571726968111992432562814401} a^{19} + \frac{890242366043241982177816102814854338787726246955575827205809806561857556673}{3021954301553833138809874076771984027243457302815571726968111992432562814401} a^{18} - \frac{121718130813257264530021480233097490175981488683983113852438961349484051980}{3021954301553833138809874076771984027243457302815571726968111992432562814401} a^{17} - \frac{1195978437864697429491778203269826781010936576135050299836464783131046823118}{3021954301553833138809874076771984027243457302815571726968111992432562814401} a^{16} - \frac{1231956759155137903431579883849054320241206079884445804437397777852576191003}{3021954301553833138809874076771984027243457302815571726968111992432562814401} a^{15} - \frac{13543541362005903882065126339754801065720337873517569524191749265304290347}{51219564433115815912031764013084475038024700047721554694374779532755301939} a^{14} - \frac{1198534784939328457240238929275988313884900427944431140341381586859632591321}{3021954301553833138809874076771984027243457302815571726968111992432562814401} a^{13} - \frac{721680920703884047080027163548557509104637897320084966265593124780681583172}{3021954301553833138809874076771984027243457302815571726968111992432562814401} a^{12} + \frac{580899171363668953203323034337946505069711102449827247108970995953797200363}{3021954301553833138809874076771984027243457302815571726968111992432562814401} a^{11} - \frac{4057382271684263398044662038652350409316005117462524137964114146017235625}{51219564433115815912031764013084475038024700047721554694374779532755301939} a^{10} - \frac{859642965208839132497024869596456434651289157973254731797723496561960163462}{3021954301553833138809874076771984027243457302815571726968111992432562814401} a^{9} - \frac{219796335244367179393320614699259749394305784782281011327445271271632656381}{3021954301553833138809874076771984027243457302815571726968111992432562814401} a^{8} + \frac{1447156711321487482792300125576802746458947048602004958545497183297783908124}{3021954301553833138809874076771984027243457302815571726968111992432562814401} a^{7} - \frac{746931531773050221014968470706203162315011418422553532764141699983020789133}{3021954301553833138809874076771984027243457302815571726968111992432562814401} a^{6} - \frac{961272264573014918517784540398450532353140182809750131076130808931479542338}{3021954301553833138809874076771984027243457302815571726968111992432562814401} a^{5} + \frac{350416712922304210882159503350073905779436176192577809367811469742085928000}{3021954301553833138809874076771984027243457302815571726968111992432562814401} a^{4} - \frac{602852206062203799476864732609206357498803538058791334308558097817575533154}{3021954301553833138809874076771984027243457302815571726968111992432562814401} a^{3} + \frac{76705063015166219321099459002390145754950837473910444070945521162504128443}{3021954301553833138809874076771984027243457302815571726968111992432562814401} a^{2} + \frac{1053224068647679910915293227298250245510566895684135410153342104734907811775}{3021954301553833138809874076771984027243457302815571726968111992432562814401} a + \frac{16766692047901009026876201207003722561189111423976956188356820855673768126}{97482396824317198026124970218451097653014751703728120224777806207502026271}$

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:  $15$
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:  \( 40139098543.6 \) (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

\(\Q(\sqrt{5}) \), 4.4.31600.1, 10.6.14339628125.1

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 R ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ $20$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{8}$

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
79Data not computed
4588681Data not computed