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Properties

Label 20.12.2821193160...1264.3
Degree $20$
Signature $[12, 4]$
Discriminant $2^{40}\cdot 11^{16}\cdot 89^{5}$
Root discriminant $83.66$
Ramified primes $2, 11, 89$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 20T310

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![399521, 0, -329600, 0, -418708, 0, 237094, 0, 57819, 0, -35350, 0, -1251, 0, 1584, 0, -59, 0, -18, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 18*x^18 - 59*x^16 + 1584*x^14 - 1251*x^12 - 35350*x^10 + 57819*x^8 + 237094*x^6 - 418708*x^4 - 329600*x^2 + 399521)
 
gp: K = bnfinit(x^20 - 18*x^18 - 59*x^16 + 1584*x^14 - 1251*x^12 - 35350*x^10 + 57819*x^8 + 237094*x^6 - 418708*x^4 - 329600*x^2 + 399521, 1)
 

Normalized defining polynomial

\( x^{20} - 18 x^{18} - 59 x^{16} + 1584 x^{14} - 1251 x^{12} - 35350 x^{10} + 57819 x^{8} + 237094 x^{6} - 418708 x^{4} - 329600 x^{2} + 399521 \)

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:  \(282119316059236304095781411736743051264=2^{40}\cdot 11^{16}\cdot 89^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $83.66$
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:  $2, 11, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{67} a^{17} - \frac{11}{67} a^{15} - \frac{2}{67} a^{13} + \frac{29}{67} a^{11} + \frac{24}{67} a^{9} - \frac{7}{67} a^{7} + \frac{16}{67} a^{5} + \frac{26}{67} a^{3} + \frac{23}{67} a$, $\frac{1}{53707493104842837922440899} a^{18} + \frac{19231422290151114250974963}{53707493104842837922440899} a^{16} + \frac{14877921862602904845811335}{53707493104842837922440899} a^{14} + \frac{25180681656656575645178307}{53707493104842837922440899} a^{12} - \frac{18861616623611801342009544}{53707493104842837922440899} a^{10} - \frac{17405868882245305493482986}{53707493104842837922440899} a^{8} + \frac{4196631775599495818181922}{53707493104842837922440899} a^{6} + \frac{24324571837921131085726391}{53707493104842837922440899} a^{4} + \frac{1097617424003925824708656}{53707493104842837922440899} a^{2} - \frac{167294279341728869248831}{801604374699146834663297}$, $\frac{1}{53707493104842837922440899} a^{19} - \frac{7082702628409780944165}{53707493104842837922440899} a^{17} + \frac{11671504363806317507158147}{53707493104842837922440899} a^{15} + \frac{9950198537372785786575664}{53707493104842837922440899} a^{13} + \frac{14004162739053218879185633}{53707493104842837922440899} a^{11} + \frac{4237449234631659042426033}{53707493104842837922440899} a^{9} - \frac{22256312589472349725706879}{53707493104842837922440899} a^{7} - \frac{14954042522337063812775162}{53707493104842837922440899} a^{5} - \frac{15736074444678157703220581}{53707493104842837922440899} a^{3} - \frac{24034386711082183594284429}{53707493104842837922440899} a$

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:  \( 604968580173 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T310:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 5120
The 44 conjugacy class representatives for t20n310
Character table for t20n310 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.10.1738687177114624.1

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 R ${\href{/LocalNumberField/3.10.0.1}{10} }{,}\,{\href{/LocalNumberField/3.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }{,}\,{\href{/LocalNumberField/13.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }^{2}$

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
$11$11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
$89$$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
89.2.1.2$x^{2} + 267$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.4.2.1$x^{4} + 979 x^{2} + 285156$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
89.4.0.1$x^{4} - x + 27$$1$$4$$0$$C_4$$[\ ]^{4}$
89.4.2.1$x^{4} + 979 x^{2} + 285156$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$