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Properties

Label 20.4.10145796540...3184.1
Degree $20$
Signature $[4, 8]$
Discriminant $2^{20}\cdot 3^{2}\cdot 401^{10}$
Root discriminant $44.70$
Ramified primes $2, 3, 401$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T347

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9, 0, 78, 0, -29, 0, -1068, 0, -1503, 0, 362, 0, 1656, 0, 976, 0, 239, 0, 26, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 26*x^18 + 239*x^16 + 976*x^14 + 1656*x^12 + 362*x^10 - 1503*x^8 - 1068*x^6 - 29*x^4 + 78*x^2 + 9)
 
gp: K = bnfinit(x^20 + 26*x^18 + 239*x^16 + 976*x^14 + 1656*x^12 + 362*x^10 - 1503*x^8 - 1068*x^6 - 29*x^4 + 78*x^2 + 9, 1)
 

Normalized defining polynomial

\( x^{20} + 26 x^{18} + 239 x^{16} + 976 x^{14} + 1656 x^{12} + 362 x^{10} - 1503 x^{8} - 1068 x^{6} - 29 x^{4} + 78 x^{2} + 9 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1014579654071904132855062602973184=2^{20}\cdot 3^{2}\cdot 401^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $44.70$
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:  $2, 3, 401$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{6} a^{12} + \frac{1}{6} a^{10} - \frac{1}{3} a^{8} - \frac{1}{6} a^{6} - \frac{1}{2} a^{4} + \frac{1}{6} a^{2}$, $\frac{1}{6} a^{13} + \frac{1}{6} a^{11} - \frac{1}{3} a^{9} - \frac{1}{6} a^{7} - \frac{1}{2} a^{5} + \frac{1}{6} a^{3}$, $\frac{1}{6} a^{14} - \frac{1}{3} a^{8} - \frac{1}{3} a^{6} + \frac{1}{6} a^{4} + \frac{1}{3} a^{2} - \frac{1}{2}$, $\frac{1}{6} a^{15} - \frac{1}{3} a^{9} - \frac{1}{3} a^{7} + \frac{1}{6} a^{5} + \frac{1}{3} a^{3} - \frac{1}{2} a$, $\frac{1}{78} a^{16} + \frac{1}{78} a^{14} + \frac{1}{78} a^{12} - \frac{8}{39} a^{10} - \frac{1}{26} a^{8} + \frac{5}{39} a^{6} - \frac{3}{26} a^{4} - \frac{3}{26} a^{2} - \frac{1}{13}$, $\frac{1}{78} a^{17} + \frac{1}{78} a^{15} + \frac{1}{78} a^{13} - \frac{8}{39} a^{11} - \frac{1}{26} a^{9} + \frac{5}{39} a^{7} - \frac{3}{26} a^{5} - \frac{3}{26} a^{3} - \frac{1}{13} a$, $\frac{1}{1674738} a^{18} + \frac{3820}{837369} a^{16} - \frac{58153}{1674738} a^{14} + \frac{20467}{1674738} a^{12} + \frac{64705}{558246} a^{10} - \frac{565501}{1674738} a^{8} + \frac{60065}{186082} a^{6} - \frac{85567}{558246} a^{4} - \frac{60713}{128826} a^{2} - \frac{127949}{558246}$, $\frac{1}{1674738} a^{19} + \frac{3820}{837369} a^{17} - \frac{58153}{1674738} a^{15} + \frac{20467}{1674738} a^{13} + \frac{64705}{558246} a^{11} - \frac{565501}{1674738} a^{9} + \frac{60065}{186082} a^{7} - \frac{85567}{558246} a^{5} - \frac{60713}{128826} a^{3} - \frac{127949}{558246} a$

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:  $11$
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:  \( 421991730.36 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T347:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 5120
The 104 conjugacy class representatives for t20n347 are not computed
Character table for t20n347 is not computed

Intermediate fields

\(\Q(\sqrt{401}) \), 5.5.160801.1 x5, 10.10.10368641602001.1

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 R ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{6}$

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}$$[2]^{5}$
2.10.10.13$x^{10} - 15 x^{8} + 26 x^{6} - 22 x^{4} + 37 x^{2} - 59$$2$$5$$10$$C_2 \times (C_2^4 : C_5)$$[2, 2, 2, 2, 2]^{5}$
$3$3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
401Data not computed