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Properties

Label 20.0.35251755556...0000.7
Degree $20$
Signature $[0, 10]$
Discriminant $2^{40}\cdot 3^{10}\cdot 5^{10}\cdot 11^{18}$
Root discriminant $134.08$
Ramified primes $2, 3, 5, 11$
Class number $4336368$ (GRH)
Class group $[2, 2, 1084092]$ (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![1181640625, 0, 4726562500, 0, 5813671875, 0, 3289687500, 0, 988796875, 0, 165550000, 0, 15290000, 0, 742500, 0, 18425, 0, 220, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 220*x^18 + 18425*x^16 + 742500*x^14 + 15290000*x^12 + 165550000*x^10 + 988796875*x^8 + 3289687500*x^6 + 5813671875*x^4 + 4726562500*x^2 + 1181640625)
 
gp: K = bnfinit(x^20 + 220*x^18 + 18425*x^16 + 742500*x^14 + 15290000*x^12 + 165550000*x^10 + 988796875*x^8 + 3289687500*x^6 + 5813671875*x^4 + 4726562500*x^2 + 1181640625, 1)
 

Normalized defining polynomial

\( x^{20} + 220 x^{18} + 18425 x^{16} + 742500 x^{14} + 15290000 x^{12} + 165550000 x^{10} + 988796875 x^{8} + 3289687500 x^{6} + 5813671875 x^{4} + 4726562500 x^{2} + 1181640625 \)

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:  \(3525175555633378160676822382018560000000000=2^{40}\cdot 3^{10}\cdot 5^{10}\cdot 11^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $134.08$
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:  $2, 3, 5, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1320=2^{3}\cdot 3\cdot 5\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{1320}(1,·)$, $\chi_{1320}(899,·)$, $\chi_{1320}(961,·)$, $\chi_{1320}(841,·)$, $\chi_{1320}(589,·)$, $\chi_{1320}(1069,·)$, $\chi_{1320}(659,·)$, $\chi_{1320}(469,·)$, $\chi_{1320}(71,·)$, $\chi_{1320}(911,·)$, $\chi_{1320}(349,·)$, $\chi_{1320}(551,·)$, $\chi_{1320}(361,·)$, $\chi_{1320}(299,·)$, $\chi_{1320}(109,·)$, $\chi_{1320}(1139,·)$, $\chi_{1320}(311,·)$, $\chi_{1320}(1081,·)$, $\chi_{1320}(1019,·)$, $\chi_{1320}(191,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{5} a^{2}$, $\frac{1}{5} a^{3}$, $\frac{1}{25} a^{4}$, $\frac{1}{25} a^{5}$, $\frac{1}{125} a^{6}$, $\frac{1}{125} a^{7}$, $\frac{1}{625} a^{8}$, $\frac{1}{625} a^{9}$, $\frac{1}{34375} a^{10}$, $\frac{1}{34375} a^{11}$, $\frac{1}{171875} a^{12}$, $\frac{1}{171875} a^{13}$, $\frac{1}{859375} a^{14}$, $\frac{1}{859375} a^{15}$, $\frac{1}{98828125} a^{16} - \frac{9}{19765625} a^{14} + \frac{2}{3953125} a^{12} - \frac{2}{790625} a^{10} + \frac{6}{14375} a^{8} - \frac{1}{575} a^{6} + \frac{11}{575} a^{4} - \frac{2}{23} a^{2} + \frac{5}{23}$, $\frac{1}{98828125} a^{17} - \frac{9}{19765625} a^{15} + \frac{2}{3953125} a^{13} - \frac{2}{790625} a^{11} + \frac{6}{14375} a^{9} - \frac{1}{575} a^{7} + \frac{11}{575} a^{5} - \frac{2}{23} a^{3} + \frac{5}{23} a$, $\frac{1}{735554509765625} a^{18} + \frac{49597}{13373718359375} a^{16} - \frac{1394229}{2674743671875} a^{14} + \frac{17857}{534948734375} a^{12} - \frac{90366}{106989746875} a^{10} + \frac{3238501}{4279589875} a^{8} + \frac{61958}{16915375} a^{6} + \frac{1085521}{77810725} a^{4} + \frac{382823}{15562145} a^{2} - \frac{1039438}{3112429}$, $\frac{1}{735554509765625} a^{19} + \frac{49597}{13373718359375} a^{17} - \frac{1394229}{2674743671875} a^{15} + \frac{17857}{534948734375} a^{13} - \frac{90366}{106989746875} a^{11} + \frac{3238501}{4279589875} a^{9} + \frac{61958}{16915375} a^{7} + \frac{1085521}{77810725} a^{5} + \frac{382823}{15562145} a^{3} - \frac{1039438}{3112429} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{1084092}$, which has order $4336368$ (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:  \( 1746210.0427691017 \) (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

\(\Q(\sqrt{-330}) \), \(\Q(\sqrt{-110}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{3}, \sqrt{-110})\), \(\Q(\zeta_{11})^+\), 10.0.58673283984691200000.1, 10.0.241453843558400000.1, 10.10.53339349076992.1

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 R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
2Data not computed
$3$3.10.5.1$x^{10} - 18 x^{6} + 81 x^{2} - 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
3.10.5.1$x^{10} - 18 x^{6} + 81 x^{2} - 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
5Data not computed
$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}$