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Properties

Label 20.0.44322362984...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{40}\cdot 3^{10}\cdot 5^{10}\cdot 31^{18}$
Root discriminant $340.67$
Ramified primes $2, 3, 5, 31$
Class number $172356096$ (GRH)
Class group $[2, 2, 2, 2, 2, 404, 13332]$ (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![961, 0, 161448, 0, 1794187, 0, 7465048, 0, 14210307, 0, 12200608, 0, 3933032, 0, 409448, 0, 16337, 0, 248, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 248*x^18 + 16337*x^16 + 409448*x^14 + 3933032*x^12 + 12200608*x^10 + 14210307*x^8 + 7465048*x^6 + 1794187*x^4 + 161448*x^2 + 961)
 
gp: K = bnfinit(x^20 + 248*x^18 + 16337*x^16 + 409448*x^14 + 3933032*x^12 + 12200608*x^10 + 14210307*x^8 + 7465048*x^6 + 1794187*x^4 + 161448*x^2 + 961, 1)
 

Normalized defining polynomial

\( x^{20} + 248 x^{18} + 16337 x^{16} + 409448 x^{14} + 3933032 x^{12} + 12200608 x^{10} + 14210307 x^{8} + 7465048 x^{6} + 1794187 x^{4} + 161448 x^{2} + 961 \)

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:  \(443223629840246396758117587996045905756160000000000=2^{40}\cdot 3^{10}\cdot 5^{10}\cdot 31^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $340.67$
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:  $2, 3, 5, 31$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3720=2^{3}\cdot 3\cdot 5\cdot 31\)
Dirichlet character group:    $\lbrace$$\chi_{3720}(1,·)$, $\chi_{3720}(709,·)$, $\chi_{3720}(1799,·)$, $\chi_{3720}(2761,·)$, $\chi_{3720}(371,·)$, $\chi_{3720}(1549,·)$, $\chi_{3720}(2509,·)$, $\chi_{3720}(721,·)$, $\chi_{3720}(2519,·)$, $\chi_{3720}(2639,·)$, $\chi_{3720}(481,·)$, $\chi_{3720}(3611,·)$, $\chi_{3720}(2279,·)$, $\chi_{3720}(1331,·)$, $\chi_{3720}(3371,·)$, $\chi_{3720}(1069,·)$, $\chi_{3720}(839,·)$, $\chi_{3720}(3251,·)$, $\chi_{3720}(841,·)$, $\chi_{3720}(829,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{5} a^{8} - \frac{2}{5} a^{4} + \frac{1}{5}$, $\frac{1}{5} a^{9} - \frac{2}{5} a^{5} + \frac{1}{5} a$, $\frac{1}{155} a^{10} - \frac{2}{5} a^{6} + \frac{1}{5} a^{2}$, $\frac{1}{155} a^{11} - \frac{2}{5} a^{7} + \frac{1}{5} a^{3}$, $\frac{1}{47275} a^{12} - \frac{24}{9455} a^{10} + \frac{37}{1525} a^{8} - \frac{92}{305} a^{6} + \frac{78}{1525} a^{4} + \frac{141}{305} a^{2} + \frac{89}{1525}$, $\frac{1}{47275} a^{13} - \frac{24}{9455} a^{11} + \frac{37}{1525} a^{9} - \frac{92}{305} a^{7} + \frac{78}{1525} a^{5} + \frac{141}{305} a^{3} + \frac{89}{1525} a$, $\frac{1}{47275} a^{14} - \frac{138}{47275} a^{10} + \frac{3}{305} a^{8} - \frac{527}{1525} a^{6} - \frac{1}{5} a^{4} + \frac{204}{1525} a^{2} + \frac{123}{305}$, $\frac{1}{47275} a^{15} - \frac{138}{47275} a^{11} + \frac{3}{305} a^{9} - \frac{527}{1525} a^{7} - \frac{1}{5} a^{5} + \frac{204}{1525} a^{3} + \frac{123}{305} a$, $\frac{1}{236375} a^{16} + \frac{1}{236375} a^{12} - \frac{2}{9455} a^{10} + \frac{41}{7625} a^{8} - \frac{81}{305} a^{6} + \frac{1896}{7625} a^{4} + \frac{138}{305} a^{2} + \frac{171}{7625}$, $\frac{1}{236375} a^{17} + \frac{1}{236375} a^{13} - \frac{2}{9455} a^{11} + \frac{41}{7625} a^{9} - \frac{81}{305} a^{7} + \frac{1896}{7625} a^{5} + \frac{138}{305} a^{3} + \frac{171}{7625} a$, $\frac{1}{8564261043288625} a^{18} - \frac{6102610618}{8564261043288625} a^{16} - \frac{63602416329}{8564261043288625} a^{14} - \frac{32739556408}{8564261043288625} a^{12} - \frac{26304531423639}{8564261043288625} a^{10} + \frac{3757649559857}{276266485267375} a^{8} + \frac{92526403203956}{276266485267375} a^{6} + \frac{127695075098802}{276266485267375} a^{4} + \frac{41044739188276}{276266485267375} a^{2} + \frac{37079681697812}{276266485267375}$, $\frac{1}{8564261043288625} a^{19} - \frac{6102610618}{8564261043288625} a^{17} - \frac{63602416329}{8564261043288625} a^{15} - \frac{32739556408}{8564261043288625} a^{13} - \frac{26304531423639}{8564261043288625} a^{11} + \frac{3757649559857}{276266485267375} a^{9} + \frac{92526403203956}{276266485267375} a^{7} + \frac{127695075098802}{276266485267375} a^{5} + \frac{41044739188276}{276266485267375} a^{3} + \frac{37079681697812}{276266485267375} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{404}\times C_{13332}$, which has order $172356096$ (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:  \( 6313916701.897883 \) (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

\(\Q(\sqrt{-310}) \), \(\Q(\sqrt{-186}) \), \(\Q(\sqrt{15}) \), \(\Q(\sqrt{15}, \sqrt{-186})\), 5.5.923521.1, 10.0.2707417309252710400000.1, 10.0.210528769967490760704.1, 10.10.663208070714121600000.1

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 ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/37.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
3Data not computed
$5$5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
31Data not computed