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Properties

Label 20.0.97243636996...0256.7
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 $16368$ (GRH)
Class group $[2, 8184]$ (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![10182481, 0, 3394120, 0, 1130970, 0, 375417, 0, 121993, 0, 36994, 0, 9662, 0, 1974, 0, 284, 0, 25, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 25*x^18 + 284*x^16 + 1974*x^14 + 9662*x^12 + 36994*x^10 + 121993*x^8 + 375417*x^6 + 1130970*x^4 + 3394120*x^2 + 10182481)
 
gp: K = bnfinit(x^20 + 25*x^18 + 284*x^16 + 1974*x^14 + 9662*x^12 + 36994*x^10 + 121993*x^8 + 375417*x^6 + 1130970*x^4 + 3394120*x^2 + 10182481, 1)
 

Normalized defining polynomial

\( x^{20} + 25 x^{18} + 284 x^{16} + 1974 x^{14} + 9662 x^{12} + 36994 x^{10} + 121993 x^{8} + 375417 x^{6} + 1130970 x^{4} + 3394120 x^{2} + 10182481 \)

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$$\chi_{924}(1,·)$, $\chi_{924}(323,·)$, $\chi_{924}(391,·)$, $\chi_{924}(841,·)$, $\chi_{924}(139,·)$, $\chi_{924}(461,·)$, $\chi_{924}(911,·)$, $\chi_{924}(71,·)$, $\chi_{924}(155,·)$, $\chi_{924}(421,·)$, $\chi_{924}(545,·)$, $\chi_{924}(475,·)$, $\chi_{924}(293,·)$, $\chi_{924}(41,·)$, $\chi_{924}(811,·)$, $\chi_{924}(307,·)$, $\chi_{924}(757,·)$, $\chi_{924}(169,·)$, $\chi_{924}(575,·)$, $\chi_{924}(629,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{3191} a^{11} + \frac{11}{3191} a^{9} + \frac{44}{3191} a^{7} + \frac{77}{3191} a^{5} + \frac{55}{3191} a^{3} + \frac{11}{3191} a$, $\frac{1}{6665999} a^{12} + \frac{1391287}{6665999} a^{10} + \frac{580806}{6665999} a^{8} + \frac{2032744}{6665999} a^{6} + \frac{2903865}{6665999} a^{4} + \frac{1451916}{6665999} a^{2} + \frac{872}{2089}$, $\frac{1}{6665999} a^{13} + \frac{13}{6665999} a^{11} - \frac{1391210}{6665999} a^{9} + \frac{810679}{6665999} a^{7} + \frac{2431751}{6665999} a^{5} - \frac{1742165}{6665999} a^{3} + \frac{810536}{6665999} a$, $\frac{1}{6665999} a^{14} + \frac{520056}{6665999} a^{10} - \frac{73800}{6665999} a^{8} + \frac{2670075}{6665999} a^{6} + \frac{503584}{6665999} a^{4} + \frac{1933625}{6665999} a^{2} - \frac{891}{2089}$, $\frac{1}{6665999} a^{15} - \frac{105}{6665999} a^{11} + \frac{870428}{6665999} a^{9} - \frac{219012}{6665999} a^{7} + \frac{447181}{6665999} a^{5} - \frac{11234}{6665999} a^{3} - \frac{1898953}{6665999} a$, $\frac{1}{6665999} a^{16} + \frac{303585}{6665999} a^{10} + \frac{771627}{6665999} a^{8} + \frac{573333}{6665999} a^{6} - \frac{1741363}{6665999} a^{4} - \frac{2765750}{6665999} a^{2} - \frac{356}{2089}$, $\frac{1}{6665999} a^{17} + \frac{680}{6665999} a^{11} - \frac{2560328}{6665999} a^{9} + \frac{577511}{6665999} a^{7} + \frac{1598948}{6665999} a^{5} + \frac{572472}{6665999} a^{3} + \frac{2198048}{6665999} a$, $\frac{1}{6665999} a^{18} - \frac{2063630}{6665999} a^{10} - \frac{1076628}{6665999} a^{8} - \frac{805179}{6665999} a^{6} - \frac{920024}{6665999} a^{4} + \frac{1463020}{6665999} a^{2} + \frac{316}{2089}$, $\frac{1}{6665999} a^{19} + \frac{302}{6665999} a^{11} + \frac{1628627}{6665999} a^{9} - \frac{3316157}{6665999} a^{7} - \frac{1981236}{6665999} a^{5} + \frac{1657297}{6665999} a^{3} - \frac{2952388}{6665999} a$

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

Class group and class number

$C_{2}\times C_{8184}$, which has order $16368$ (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:  \( 1746210.04277 \) (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

\(\Q(\sqrt{-77}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-231}) \), \(\Q(\sqrt{3}, \sqrt{-77})\), \(\Q(\zeta_{11})^+\), 10.0.40581147486860288.1, 10.10.53339349076992.1, 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 ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ R R ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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}$
$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}$
7Data 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}$