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Properties

Label 20.0.35251755556...0000.3
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 $1641760$ (GRH)
Class group $[2, 2, 410440]$ (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![31236921, 0, 14062322520, 0, 15157215639, 0, 6266008692, 0, 1295855091, 0, 151861392, 0, 10700100, 0, 461268, 0, 11889, 0, 168, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 168*x^18 + 11889*x^16 + 461268*x^14 + 10700100*x^12 + 151861392*x^10 + 1295855091*x^8 + 6266008692*x^6 + 15157215639*x^4 + 14062322520*x^2 + 31236921)
 
gp: K = bnfinit(x^20 + 168*x^18 + 11889*x^16 + 461268*x^14 + 10700100*x^12 + 151861392*x^10 + 1295855091*x^8 + 6266008692*x^6 + 15157215639*x^4 + 14062322520*x^2 + 31236921, 1)
 

Normalized defining polynomial

\( x^{20} + 168 x^{18} + 11889 x^{16} + 461268 x^{14} + 10700100 x^{12} + 151861392 x^{10} + 1295855091 x^{8} + 6266008692 x^{6} + 15157215639 x^{4} + 14062322520 x^{2} + 31236921 \)

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$$\chi_{1320}(1,·)$, $\chi_{1320}(131,·)$, $\chi_{1320}(389,·)$, $\chi_{1320}(961,·)$, $\chi_{1320}(841,·)$, $\chi_{1320}(269,·)$, $\chi_{1320}(79,·)$, $\chi_{1320}(1091,·)$, $\chi_{1320}(1109,·)$, $\chi_{1320}(919,·)$, $\chi_{1320}(799,·)$, $\chi_{1320}(611,·)$, $\chi_{1320}(679,·)$, $\chi_{1320}(361,·)$, $\chi_{1320}(491,·)$, $\chi_{1320}(749,·)$, $\chi_{1320}(371,·)$, $\chi_{1320}(439,·)$, $\chi_{1320}(1081,·)$, $\chi_{1320}(509,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{3} a^{2}$, $\frac{1}{3} a^{3}$, $\frac{1}{9} a^{4}$, $\frac{1}{9} a^{5}$, $\frac{1}{27} a^{6}$, $\frac{1}{27} a^{7}$, $\frac{1}{81} a^{8}$, $\frac{1}{81} a^{9}$, $\frac{1}{243} a^{10}$, $\frac{1}{5589} a^{11} + \frac{10}{1863} a^{9} + \frac{5}{621} a^{7} + \frac{1}{23} a^{5} - \frac{7}{69} a^{3} + \frac{5}{23} a$, $\frac{1}{9942831} a^{12} - \frac{4705}{3314277} a^{10} - \frac{5354}{1104759} a^{8} - \frac{3740}{368253} a^{6} + \frac{4846}{122751} a^{4} - \frac{3560}{40917} a^{2} + \frac{101}{593}$, $\frac{1}{9942831} a^{13} + \frac{13}{1104759} a^{11} + \frac{1169}{1104759} a^{9} + \frac{6341}{368253} a^{7} + \frac{6625}{122751} a^{5} + \frac{1383}{13639} a^{3} - \frac{1235}{13639} a$, $\frac{1}{29828493} a^{14} - \frac{698}{368253} a^{10} - \frac{3077}{1104759} a^{8} + \frac{2456}{368253} a^{6} + \frac{6101}{122751} a^{4} + \frac{405}{13639} a^{2} + \frac{212}{593}$, $\frac{1}{29828493} a^{15} + \frac{241}{3314277} a^{11} - \frac{2014}{368253} a^{9} - \frac{5846}{368253} a^{7} - \frac{1129}{40917} a^{5} - \frac{3529}{40917} a^{3} - \frac{3426}{13639} a$, $\frac{1}{89485479} a^{16} - \frac{4174}{3314277} a^{10} + \frac{2402}{1104759} a^{8} - \frac{2221}{368253} a^{6} + \frac{171}{13639} a^{4} - \frac{4723}{40917} a^{2} - \frac{28}{593}$, $\frac{1}{89485479} a^{17} - \frac{1}{144099} a^{11} + \frac{2995}{1104759} a^{9} + \frac{4895}{368253} a^{7} - \frac{673}{40917} a^{5} - \frac{6502}{40917} a^{3} + \frac{6472}{13639} a$, $\frac{1}{268456437} a^{18} + \frac{3892}{3314277} a^{10} + \frac{4504}{1104759} a^{8} - \frac{6205}{368253} a^{6} - \frac{4156}{122751} a^{4} + \frac{2142}{13639} a^{2} - \frac{49}{593}$, $\frac{1}{268456437} a^{19} - \frac{259}{3314277} a^{11} + \frac{3911}{1104759} a^{9} + \frac{106}{122751} a^{7} - \frac{26}{5337} a^{5} - \frac{5434}{40917} a^{3} + \frac{5396}{13639} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{410440}$, which has order $1641760$ (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:  \( 11184526.893275889 \) (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

\(\Q(\sqrt{-66}) \), \(\Q(\sqrt{-30}) \), \(\Q(\sqrt{55}) \), \(\Q(\sqrt{-30}, \sqrt{55})\), \(\Q(\zeta_{11})^+\), 10.0.18775450875101184.1, 10.0.5333934907699200000.1, 10.10.7545432611200000.1

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 ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.1.0.1}{1} }^{20}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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$3.10.5.2$x^{10} - 81 x^{2} + 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
3.10.5.2$x^{10} - 81 x^{2} + 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$5$5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$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}$