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Properties

Label 20.0.18218737052...0000.2
Degree $20$
Signature $[0, 10]$
Discriminant $2^{30}\cdot 5^{15}\cdot 11^{18}$
Root discriminant $81.85$
Ramified primes $2, 5, 11$
Class number $80$ (GRH)
Class group $[2, 2, 20]$ (GRH)
Galois group $C_2\times F_5$ (as 20T9)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![247808000, 0, -179660800, 0, 63222016, 0, -2787840, 0, -727936, 0, 330880, 0, -27984, 0, 3520, 0, 264, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 264*x^16 + 3520*x^14 - 27984*x^12 + 330880*x^10 - 727936*x^8 - 2787840*x^6 + 63222016*x^4 - 179660800*x^2 + 247808000)
 
gp: K = bnfinit(x^20 + 264*x^16 + 3520*x^14 - 27984*x^12 + 330880*x^10 - 727936*x^8 - 2787840*x^6 + 63222016*x^4 - 179660800*x^2 + 247808000, 1)
 

Normalized defining polynomial

\( x^{20} + 264 x^{16} + 3520 x^{14} - 27984 x^{12} + 330880 x^{10} - 727936 x^{8} - 2787840 x^{6} + 63222016 x^{4} - 179660800 x^{2} + 247808000 \)

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:  \(182187370528513441169408000000000000000=2^{30}\cdot 5^{15}\cdot 11^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $81.85$
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:  $2, 5, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{4} a^{4}$, $\frac{1}{8} a^{5} - \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{32} a^{6} - \frac{1}{16} a^{4} + \frac{1}{8} a^{2}$, $\frac{1}{32} a^{7} - \frac{1}{16} a^{5} + \frac{1}{8} a^{3}$, $\frac{1}{64} a^{8} + \frac{1}{8} a^{2}$, $\frac{1}{64} a^{9} + \frac{1}{8} a^{3}$, $\frac{1}{1408} a^{10} - \frac{1}{16} a^{4}$, $\frac{1}{2816} a^{11} - \frac{1}{64} a^{7} + \frac{3}{16} a^{3} - \frac{1}{2} a$, $\frac{1}{11264} a^{12} - \frac{1}{2816} a^{10} + \frac{1}{256} a^{8} - \frac{1}{64} a^{6} - \frac{1}{64} a^{4} - \frac{1}{8} a^{2}$, $\frac{1}{11264} a^{13} + \frac{1}{256} a^{9} + \frac{3}{64} a^{5} - \frac{1}{16} a^{3}$, $\frac{1}{22528} a^{14} - \frac{1}{5632} a^{10} - \frac{1}{128} a^{6} - \frac{1}{32} a^{4} - \frac{1}{8} a^{2}$, $\frac{1}{22528} a^{15} - \frac{1}{5632} a^{11} - \frac{1}{128} a^{7} - \frac{1}{32} a^{5} - \frac{1}{8} a^{3}$, $\frac{1}{10633216} a^{16} - \frac{47}{5316608} a^{14} - \frac{51}{1329152} a^{12} - \frac{103}{1329152} a^{10} - \frac{81}{30208} a^{8} - \frac{169}{30208} a^{6} - \frac{721}{15104} a^{4} - \frac{391}{1888} a^{2} + \frac{67}{236}$, $\frac{1}{10633216} a^{17} - \frac{47}{5316608} a^{15} - \frac{51}{1329152} a^{13} - \frac{103}{1329152} a^{11} - \frac{81}{30208} a^{9} - \frac{169}{30208} a^{7} - \frac{721}{15104} a^{5} - \frac{391}{1888} a^{3} + \frac{67}{236} a$, $\frac{1}{7897860582298255360} a^{18} - \frac{1287147545}{71798732566347776} a^{16} + \frac{1309322656123}{89748415707934720} a^{14} + \frac{608536983891}{17949683141586944} a^{12} + \frac{1917818307263}{22437103926983680} a^{10} - \frac{11450845237705}{4487420785396736} a^{8} - \frac{7033951533139}{1019868360317440} a^{6} - \frac{1449328826033}{12748354503968} a^{4} + \frac{2980018942931}{15935443129960} a^{2} + \frac{189290957758}{398386078249}$, $\frac{1}{39489302911491276800} a^{19} - \frac{257429509}{71798732566347776} a^{17} + \frac{9277044221103}{448742078539673600} a^{15} + \frac{3795625609883}{89748415707934720} a^{13} - \frac{549991205247}{10198683603174400} a^{11} - \frac{46508820123617}{22437103926983680} a^{9} - \frac{54840280923019}{5099341801587200} a^{7} - \frac{923857452141}{31870886259920} a^{5} + \frac{621493666772}{9959651956225} a^{3} + \frac{155393598753}{796772156498} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{20}$, which has order $80$ (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:  \( 1640482329.2124414 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times F_5$ (as 20T9):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 40
The 10 conjugacy class representatives for $C_2\times F_5$
Character table for $C_2\times F_5$

Intermediate fields

\(\Q(\sqrt{5}) \), 4.0.968000.5, 5.1.1830125.1, 10.2.16746787578125.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Galois closure: data not computed
Degree 10 siblings: data not computed
Degree 20 sibling: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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$2.4.6.4$x^{4} - 2 x^{2} + 20$$2$$2$$6$$C_4$$[3]^{2}$
2.8.12.2$x^{8} + 2 x^{6} + 8 x^{4} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
2.8.12.2$x^{8} + 2 x^{6} + 8 x^{4} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
$5$5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$11$11.10.9.8$x^{10} + 33$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.8$x^{10} + 33$$10$$1$$9$$C_{10}$$[\ ]_{10}$