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Properties

Label 20.0.97243636996...256.10
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 3^{10}\cdot 7^{10}\cdot 11^{18}$
Root discriminant $79.32$
Ramified primes $2, 3, 7, 11$
Class number $133320$ (GRH)
Class group $[2, 2, 33330]$ (GRH)
Galois group $C_2\times C_{10}$ (as 20T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![310499641, 0, 372701157, 0, 200468262, 0, 63303777, 0, 12892432, 0, 1748264, 0, 157854, 0, 9564, 0, 460, 0, 25, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 25*x^18 + 460*x^16 + 9564*x^14 + 157854*x^12 + 1748264*x^10 + 12892432*x^8 + 63303777*x^6 + 200468262*x^4 + 372701157*x^2 + 310499641)
 
gp: K = bnfinit(x^20 + 25*x^18 + 460*x^16 + 9564*x^14 + 157854*x^12 + 1748264*x^10 + 12892432*x^8 + 63303777*x^6 + 200468262*x^4 + 372701157*x^2 + 310499641, 1)
 

Normalized defining polynomial

\( x^{20} + 25 x^{18} + 460 x^{16} + 9564 x^{14} + 157854 x^{12} + 1748264 x^{10} + 12892432 x^{8} + 63303777 x^{6} + 200468262 x^{4} + 372701157 x^{2} + 310499641 \)

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:  \(97243636996704282094565176844674400256=2^{20}\cdot 3^{10}\cdot 7^{10}\cdot 11^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $79.32$
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:  $2, 3, 7, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(924=2^{2}\cdot 3\cdot 7\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{924}(799,·)$, $\chi_{924}(1,·)$, $\chi_{924}(839,·)$, $\chi_{924}(841,·)$, $\chi_{924}(587,·)$, $\chi_{924}(461,·)$, $\chi_{924}(335,·)$, $\chi_{924}(547,·)$, $\chi_{924}(251,·)$, $\chi_{924}(421,·)$, $\chi_{924}(545,·)$, $\chi_{924}(419,·)$, $\chi_{924}(293,·)$, $\chi_{924}(41,·)$, $\chi_{924}(43,·)$, $\chi_{924}(211,·)$, $\chi_{924}(757,·)$, $\chi_{924}(169,·)$, $\chi_{924}(127,·)$, $\chi_{924}(629,·)$$\rbrace$
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{67} a^{17} + \frac{19}{67} a^{15} + \frac{11}{67} a^{13} - \frac{16}{67} a^{11} + \frac{31}{67} a^{9} - \frac{19}{67} a^{7} + \frac{4}{67} a^{5} + \frac{9}{67} a^{3} - \frac{13}{67} a$, $\frac{1}{17168720944013682994845536401} a^{18} + \frac{5809089141519580897259688813}{17168720944013682994845536401} a^{16} - \frac{7826544064233169825886875178}{17168720944013682994845536401} a^{14} - \frac{1783485828382867736230466424}{17168720944013682994845536401} a^{12} - \frac{5283925219730378710252577903}{17168720944013682994845536401} a^{10} - \frac{2627669867100231147637841978}{17168720944013682994845536401} a^{8} - \frac{4018662926050154857597054751}{17168720944013682994845536401} a^{6} + \frac{7918894970054929766253297554}{17168720944013682994845536401} a^{4} + \frac{80056730618974688773523377}{192906976899030146009500409} a^{2} + \frac{79357468962059850016762440}{256249566328562432758888603}$, $\frac{1}{4515373608275598627644376073463} a^{19} - \frac{15459624863751101021728065236}{4515373608275598627644376073463} a^{17} + \frac{1201927658572910075228826219585}{4515373608275598627644376073463} a^{15} + \frac{674202870146364829881717668290}{4515373608275598627644376073463} a^{13} + \frac{1313632592673380462699747061738}{4515373608275598627644376073463} a^{11} + \frac{814552197154685366920457912989}{4515373608275598627644376073463} a^{9} - \frac{1728834493883603889757676241544}{4515373608275598627644376073463} a^{7} + \frac{1433691482022176305636709484646}{4515373608275598627644376073463} a^{5} + \frac{13554753027446752051825193737}{50734534924444928400498607567} a^{3} - \frac{1692592676072596669509272799998}{4515373608275598627644376073463} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{33330}$, which has order $133320$ (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:  $9$
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:  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:  \( 281202.490766 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_{10}$ (as 20T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 20
The 20 conjugacy class representatives for $C_2\times C_{10}$
Character table for $C_2\times C_{10}$

Intermediate fields

\(\Q(\sqrt{-21}) \), \(\Q(\sqrt{11}) \), \(\Q(\sqrt{-231}) \), \(\Q(\sqrt{11}, \sqrt{-21})\), \(\Q(\zeta_{11})^+\), 10.0.896474439937004544.3, \(\Q(\zeta_{44})^+\), 10.0.9630096522760791.1

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

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 ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ R R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$2$2.10.10.11$x^{10} - x^{8} + 3 x^{6} + x^{2} - 3$$2$$5$$10$$C_{10}$$[2]^{5}$
2.10.10.11$x^{10} - x^{8} + 3 x^{6} + x^{2} - 3$$2$$5$$10$$C_{10}$$[2]^{5}$
3Data not computed
$7$7.10.5.1$x^{10} - 98 x^{6} + 2401 x^{2} - 268912$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
7.10.5.1$x^{10} - 98 x^{6} + 2401 x^{2} - 268912$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$11$11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$