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Properties

Label 20.10.5903156224...0000.1
Degree $20$
Signature $[10, 5]$
Discriminant $-\,2^{32}\cdot 5^{22}\cdot 7^{8}$
Root discriminant $38.78$
Ramified primes $2, 5, 7$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T513

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 0, 5, 0, 115, 0, -350, 0, -3045, 0, 671, 0, 765, 0, -110, 0, -55, 0, 5, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 5*x^18 - 55*x^16 - 110*x^14 + 765*x^12 + 671*x^10 - 3045*x^8 - 350*x^6 + 115*x^4 + 5*x^2 - 1)
 
gp: K = bnfinit(x^20 + 5*x^18 - 55*x^16 - 110*x^14 + 765*x^12 + 671*x^10 - 3045*x^8 - 350*x^6 + 115*x^4 + 5*x^2 - 1, 1)
 

Normalized defining polynomial

\( x^{20} + 5 x^{18} - 55 x^{16} - 110 x^{14} + 765 x^{12} + 671 x^{10} - 3045 x^{8} - 350 x^{6} + 115 x^{4} + 5 x^{2} - 1 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-59031562240000000000000000000000=-\,2^{32}\cdot 5^{22}\cdot 7^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.78$
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:  $2, 5, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{12} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{13} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{4} a^{15} - \frac{1}{4} a^{13} - \frac{1}{4} a^{12} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} + \frac{1}{4}$, $\frac{1}{164} a^{16} + \frac{1}{82} a^{14} - \frac{1}{41} a^{12} + \frac{4}{41} a^{10} - \frac{3}{164} a^{8} - \frac{12}{41} a^{6} + \frac{11}{41} a^{4} + \frac{31}{82} a^{2} - \frac{23}{164}$, $\frac{1}{164} a^{17} + \frac{1}{82} a^{15} - \frac{1}{41} a^{13} + \frac{4}{41} a^{11} - \frac{3}{164} a^{9} - \frac{12}{41} a^{7} + \frac{11}{41} a^{5} + \frac{31}{82} a^{3} - \frac{23}{164} a$, $\frac{1}{86188724} a^{18} - \frac{38239}{86188724} a^{16} + \frac{9816937}{86188724} a^{14} - \frac{1}{4} a^{13} - \frac{4681347}{21547181} a^{12} - \frac{1}{4} a^{11} + \frac{1455332}{21547181} a^{10} + \frac{1244923}{43094362} a^{8} - \frac{1}{4} a^{7} - \frac{20972401}{86188724} a^{6} - \frac{27411395}{86188724} a^{4} - \frac{1}{4} a^{3} - \frac{40409289}{86188724} a^{2} + \frac{1}{4} a - \frac{4653925}{43094362}$, $\frac{1}{86188724} a^{19} - \frac{38239}{86188724} a^{17} + \frac{9816937}{86188724} a^{15} + \frac{2821793}{86188724} a^{13} - \frac{1}{4} a^{12} - \frac{15725853}{86188724} a^{11} - \frac{1}{4} a^{10} + \frac{1244923}{43094362} a^{9} - \frac{21259791}{43094362} a^{7} - \frac{1}{4} a^{6} - \frac{27411395}{86188724} a^{5} + \frac{12116127}{43094362} a^{3} - \frac{1}{4} a^{2} - \frac{30855031}{86188724} a + \frac{1}{4}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $14$
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:  \( 308636165.52 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T513:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 20480
The 128 conjugacy class representatives for t20n513 are not computed
Character table for t20n513 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.2450000.1, 10.10.30012500000000.1

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

Sibling fields

Degree 20 siblings: data not computed
Degree 40 siblings: 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 R ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }^{2}$

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
5Data not computed
$7$7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$