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Properties

Label 20.8.16075758655...0000.2
Degree $20$
Signature $[8, 6]$
Discriminant $2^{16}\cdot 5^{11}\cdot 3469^{5}$
Root discriminant $32.38$
Ramified primes $2, 5, 3469$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T771

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4, 4, -296, 1588, -3660, 3358, 1874, -7762, 8464, -5208, 1753, 677, -1433, 894, -343, 79, 37, -40, 13, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 13*x^18 - 40*x^17 + 37*x^16 + 79*x^15 - 343*x^14 + 894*x^13 - 1433*x^12 + 677*x^11 + 1753*x^10 - 5208*x^9 + 8464*x^8 - 7762*x^7 + 1874*x^6 + 3358*x^5 - 3660*x^4 + 1588*x^3 - 296*x^2 + 4*x + 4)
 
gp: K = bnfinit(x^20 - 5*x^19 + 13*x^18 - 40*x^17 + 37*x^16 + 79*x^15 - 343*x^14 + 894*x^13 - 1433*x^12 + 677*x^11 + 1753*x^10 - 5208*x^9 + 8464*x^8 - 7762*x^7 + 1874*x^6 + 3358*x^5 - 3660*x^4 + 1588*x^3 - 296*x^2 + 4*x + 4, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} + 13 x^{18} - 40 x^{17} + 37 x^{16} + 79 x^{15} - 343 x^{14} + 894 x^{13} - 1433 x^{12} + 677 x^{11} + 1753 x^{10} - 5208 x^{9} + 8464 x^{8} - 7762 x^{7} + 1874 x^{6} + 3358 x^{5} - 3660 x^{4} + 1588 x^{3} - 296 x^{2} + 4 x + 4 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1607575865553562716800000000000=2^{16}\cdot 5^{11}\cdot 3469^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $32.38$
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:  $2, 5, 3469$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{7}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{8}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{5}$, $\frac{1}{4} a^{18} - \frac{1}{4} a^{14} - \frac{1}{4} a^{10} - \frac{1}{2} a^{6}$, $\frac{1}{32436767277937639820924452} a^{19} - \frac{100699658277752640283540}{8109191819484409955231113} a^{18} - \frac{1560170291896910916021693}{8109191819484409955231113} a^{17} + \frac{209740701724806044312308}{8109191819484409955231113} a^{16} - \frac{719267665014708073369703}{32436767277937639820924452} a^{15} - \frac{1603061159317383563781625}{8109191819484409955231113} a^{14} - \frac{2414993348283506635202583}{16218383638968819910462226} a^{13} - \frac{820446425712255372865509}{8109191819484409955231113} a^{12} + \frac{2108772848631704710795089}{32436767277937639820924452} a^{11} + \frac{776709000379477495617792}{8109191819484409955231113} a^{10} + \frac{592914057647850942540872}{8109191819484409955231113} a^{9} - \frac{3429958625731958350831828}{8109191819484409955231113} a^{8} - \frac{5995522564655068479755147}{16218383638968819910462226} a^{7} + \frac{2851378851760075641827151}{8109191819484409955231113} a^{6} - \frac{3016889807603416378293489}{8109191819484409955231113} a^{5} + \frac{744687793964470149195906}{8109191819484409955231113} a^{4} - \frac{1573146896415777064603822}{8109191819484409955231113} a^{3} + \frac{1083260057528596938864034}{8109191819484409955231113} a^{2} - \frac{3266456535031109273424258}{8109191819484409955231113} a + \frac{2760567540391000661169194}{8109191819484409955231113}$

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:  $13$
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:  \( 51510058.233 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T771:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 102400
The 130 conjugacy class representatives for t20n771 are not computed
Character table for t20n771 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.10.9627168800000.1

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 $20$ R $20$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{3}$ $20$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ $20$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{3}$ $20$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{5}$

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$2.10.8.1$x^{10} - 2 x^{5} + 4$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
2.10.8.1$x^{10} - 2 x^{5} + 4$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3469Data not computed