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Properties

Label 20.10.4717154733...3971.3
Degree $20$
Signature $[10, 5]$
Discriminant $-\,11^{17}\cdot 1451^{6}$
Root discriminant $68.18$
Ramified primes $11, 1451$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T326

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-2333, -15797, 80270, 549299, 1089529, 772487, -274432, -661698, -229286, 56516, 46166, 3289, -15774, -3205, 4374, -24, -519, 124, 14, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 14*x^18 + 124*x^17 - 519*x^16 - 24*x^15 + 4374*x^14 - 3205*x^13 - 15774*x^12 + 3289*x^11 + 46166*x^10 + 56516*x^9 - 229286*x^8 - 661698*x^7 - 274432*x^6 + 772487*x^5 + 1089529*x^4 + 549299*x^3 + 80270*x^2 - 15797*x - 2333)
 
gp: K = bnfinit(x^20 - 10*x^19 + 14*x^18 + 124*x^17 - 519*x^16 - 24*x^15 + 4374*x^14 - 3205*x^13 - 15774*x^12 + 3289*x^11 + 46166*x^10 + 56516*x^9 - 229286*x^8 - 661698*x^7 - 274432*x^6 + 772487*x^5 + 1089529*x^4 + 549299*x^3 + 80270*x^2 - 15797*x - 2333, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 14 x^{18} + 124 x^{17} - 519 x^{16} - 24 x^{15} + 4374 x^{14} - 3205 x^{13} - 15774 x^{12} + 3289 x^{11} + 46166 x^{10} + 56516 x^{9} - 229286 x^{8} - 661698 x^{7} - 274432 x^{6} + 772487 x^{5} + 1089529 x^{4} + 549299 x^{3} + 80270 x^{2} - 15797 x - 2333 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-4717154733296640478520170220486763971=-\,11^{17}\cdot 1451^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $68.18$
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:  $11, 1451$
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}{8194100314010937526649680082150996757773509687556592962069} a^{19} + \frac{2508649538828747074675102248852475718872557725593080926855}{8194100314010937526649680082150996757773509687556592962069} a^{18} + \frac{3157786092677640006851746750075874848590878619458535770012}{8194100314010937526649680082150996757773509687556592962069} a^{17} - \frac{761213075417669914442933567475797795628834051038404744986}{8194100314010937526649680082150996757773509687556592962069} a^{16} + \frac{2703533829494302794778381096786847920789108511314884549140}{8194100314010937526649680082150996757773509687556592962069} a^{15} - \frac{1461162520881031306335176181220744908019748750293585034001}{8194100314010937526649680082150996757773509687556592962069} a^{14} - \frac{2719106282640793551163615213370330373022316847060823170104}{8194100314010937526649680082150996757773509687556592962069} a^{13} - \frac{1122037022575307979967241132230093924520811896393176249446}{8194100314010937526649680082150996757773509687556592962069} a^{12} + \frac{1902848912992520891050670082012994665654913122568533607136}{8194100314010937526649680082150996757773509687556592962069} a^{11} + \frac{2792570337933323602045369651071769136224891711653630500064}{8194100314010937526649680082150996757773509687556592962069} a^{10} - \frac{804991851306077517697726046456543210136438804364752163162}{8194100314010937526649680082150996757773509687556592962069} a^{9} - \frac{1304580538787916533906538617246657882314018910350683843332}{8194100314010937526649680082150996757773509687556592962069} a^{8} + \frac{2949619843355974642441263462844804219993718487168923040413}{8194100314010937526649680082150996757773509687556592962069} a^{7} + \frac{4047014674078999572567866415891244740975592980590436843942}{8194100314010937526649680082150996757773509687556592962069} a^{6} - \frac{1630184665142347527097809359386420498873534014436063582442}{8194100314010937526649680082150996757773509687556592962069} a^{5} + \frac{990939683559788845317151243109545327328074174299459064472}{8194100314010937526649680082150996757773509687556592962069} a^{4} + \frac{396696830953163279962008628511959319147102950161827690170}{8194100314010937526649680082150996757773509687556592962069} a^{3} + \frac{3336459610595252095152038360721286128142321155775277760204}{8194100314010937526649680082150996757773509687556592962069} a^{2} + \frac{208117697917054759765032768870209779650224898736023028097}{8194100314010937526649680082150996757773509687556592962069} a + \frac{3249593483295195402595194575418020096882506458113053693052}{8194100314010937526649680082150996757773509687556592962069}$

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:  $14$
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:  \( 90982181599.0 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T326:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 5120
The 80 conjugacy class representatives for t20n326 are not computed
Character table for t20n326 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.10.451311402416281.1

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 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 ${\href{/LocalNumberField/2.10.0.1}{10} }{,}\,{\href{/LocalNumberField/2.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }{,}\,{\href{/LocalNumberField/13.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$11$11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.10.9.7$x^{10} + 2673$$10$$1$$9$$C_{10}$$[\ ]_{10}$
1451Data not computed