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Properties

Label 20.0.16863534401...0000.6
Degree $20$
Signature $[0, 10]$
Discriminant $2^{40}\cdot 5^{10}\cdot 7^{10}\cdot 11^{18}$
Root discriminant $204.81$
Ramified primes $2, 5, 7, 11$
Class number $37284544$ (GRH)
Class group $[2, 4, 4660568]$ (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![68780832121, 0, 6878083200, 0, 687808199, 0, 68780348, 0, 6877091, 0, 686608, 0, 67860, 0, 6412, 0, 529, 0, 32, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 32*x^18 + 529*x^16 + 6412*x^14 + 67860*x^12 + 686608*x^10 + 6877091*x^8 + 68780348*x^6 + 687808199*x^4 + 6878083200*x^2 + 68780832121)
 
gp: K = bnfinit(x^20 + 32*x^18 + 529*x^16 + 6412*x^14 + 67860*x^12 + 686608*x^10 + 6877091*x^8 + 68780348*x^6 + 687808199*x^4 + 6878083200*x^2 + 68780832121, 1)
 

Normalized defining polynomial

\( x^{20} + 32 x^{18} + 529 x^{16} + 6412 x^{14} + 67860 x^{12} + 686608 x^{10} + 6877091 x^{8} + 68780348 x^{6} + 687808199 x^{4} + 6878083200 x^{2} + 68780832121 \)

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:  \(16863534401027144382603387201975746560000000000=2^{40}\cdot 5^{10}\cdot 7^{10}\cdot 11^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $204.81$
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:  $2, 5, 7, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3080=2^{3}\cdot 5\cdot 7\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{3080}(1,·)$, $\chi_{3080}(1091,·)$, $\chi_{3080}(1989,·)$, $\chi_{3080}(3079,·)$, $\chi_{3080}(841,·)$, $\chi_{3080}(1931,·)$, $\chi_{3080}(1679,·)$, $\chi_{3080}(1681,·)$, $\chi_{3080}(2771,·)$, $\chi_{3080}(1371,·)$, $\chi_{3080}(2269,·)$, $\chi_{3080}(1119,·)$, $\chi_{3080}(1961,·)$, $\chi_{3080}(811,·)$, $\chi_{3080}(1709,·)$, $\chi_{3080}(309,·)$, $\chi_{3080}(1399,·)$, $\chi_{3080}(1401,·)$, $\chi_{3080}(1149,·)$, $\chi_{3080}(2239,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{262261} a^{11} + \frac{11}{262261} a^{9} + \frac{44}{262261} a^{7} + \frac{77}{262261} a^{5} + \frac{55}{262261} a^{3} + \frac{11}{262261} a$, $\frac{1}{58130412911} a^{12} + \frac{4878316872}{58130412911} a^{10} - \frac{9347244257}{58130412911} a^{8} - \frac{3650148521}{58130412911} a^{6} + \frac{11394191461}{58130412911} a^{4} + \frac{5697095714}{58130412911} a^{2} + \frac{37202}{221651}$, $\frac{1}{58130412911} a^{13} + \frac{13}{58130412911} a^{11} - \frac{4878316795}{58130412911} a^{9} + \frac{14225561327}{58130412911} a^{7} - \frac{15453729216}{58130412911} a^{5} + \frac{28041733024}{58130412911} a^{3} + \frac{14225561184}{58130412911} a$, $\frac{1}{58130412911} a^{14} - \frac{10166023220}{58130412911} a^{10} + \frac{19478910846}{58130412911} a^{8} - \frac{26132211354}{58130412911} a^{6} - \frac{3821930147}{58130412911} a^{4} - \frac{1706270187}{58130412911} a^{2} - \frac{40324}{221651}$, $\frac{1}{58130412911} a^{15} - \frac{105}{58130412911} a^{11} + \frac{15044339289}{58130412911} a^{9} + \frac{14259915329}{58130412911} a^{7} + \frac{23266481865}{58130412911} a^{5} - \frac{23879127972}{58130412911} a^{3} - \frac{15009984121}{58130412911} a$, $\frac{1}{58130412911} a^{16} + \frac{4093894650}{58130412911} a^{10} + \frac{21016287831}{58130412911} a^{8} - \frac{11216635374}{58130412911} a^{6} + \frac{9902717213}{58130412911} a^{4} + \frac{1880936739}{58130412911} a^{2} - \frac{83508}{221651}$, $\frac{1}{58130412911} a^{17} + \frac{680}{58130412911} a^{11} - \frac{24016545839}{58130412911} a^{9} - \frac{16956731321}{58130412911} a^{7} - \frac{14675053922}{58130412911} a^{5} + \frac{9238420033}{58130412911} a^{3} - \frac{8803312347}{58130412911} a$, $\frac{1}{58130412911} a^{18} - \frac{27838482872}{58130412911} a^{10} + \frac{2954356140}{58130412911} a^{8} + \frac{25948598096}{58130412911} a^{6} - \frac{7466856284}{58130412911} a^{4} + \frac{11909267170}{58130412911} a^{2} - \frac{29146}{221651}$, $\frac{1}{58130412911} a^{19} - \frac{3876}{58130412911} a^{11} + \frac{18525560541}{58130412911} a^{9} - \frac{28027410122}{58130412911} a^{7} - \frac{14729251299}{58130412911} a^{5} - \frac{26495536647}{58130412911} a^{3} + \frac{7927345295}{58130412911} a$

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

Class group and class number

$C_{2}\times C_{4}\times C_{4660568}$, which has order $37284544$ (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:  \( 14452469.589232503 \) (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

\(\Q(\sqrt{-385}) \), \(\Q(\sqrt{-154}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{10}, \sqrt{-154})\), \(\Q(\zeta_{11})^+\), 10.0.126816085896438400000.1, 10.0.1298596719579529216.1, 10.10.21950349414400000.1

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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R R R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$5$5.10.5.2$x^{10} - 625 x^{2} + 6250$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
5.10.5.2$x^{10} - 625 x^{2} + 6250$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
7Data not computed
11Data not computed