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Properties

Label 20.12.7414590328...0000.1
Degree $20$
Signature $[12, 4]$
Discriminant $2^{20}\cdot 5^{11}\cdot 3469^{4}$
Root discriminant $24.75$
Ramified primes $2, 5, 3469$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T755

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -20, -41, 226, 99, -546, -252, 456, 659, -80, -771, -178, 393, 270, -124, -154, 33, 42, -9, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 - 9*x^18 + 42*x^17 + 33*x^16 - 154*x^15 - 124*x^14 + 270*x^13 + 393*x^12 - 178*x^11 - 771*x^10 - 80*x^9 + 659*x^8 + 456*x^7 - 252*x^6 - 546*x^5 + 99*x^4 + 226*x^3 - 41*x^2 - 20*x - 1)
 
gp: K = bnfinit(x^20 - 4*x^19 - 9*x^18 + 42*x^17 + 33*x^16 - 154*x^15 - 124*x^14 + 270*x^13 + 393*x^12 - 178*x^11 - 771*x^10 - 80*x^9 + 659*x^8 + 456*x^7 - 252*x^6 - 546*x^5 + 99*x^4 + 226*x^3 - 41*x^2 - 20*x - 1, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} - 9 x^{18} + 42 x^{17} + 33 x^{16} - 154 x^{15} - 124 x^{14} + 270 x^{13} + 393 x^{12} - 178 x^{11} - 771 x^{10} - 80 x^{9} + 659 x^{8} + 456 x^{7} - 252 x^{6} - 546 x^{5} + 99 x^{4} + 226 x^{3} - 41 x^{2} - 20 x - 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(7414590328295475200000000000=2^{20}\cdot 5^{11}\cdot 3469^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $24.75$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{5} a^{18} - \frac{1}{5} a^{17} + \frac{2}{5} a^{16} - \frac{1}{5} a^{15} - \frac{2}{5} a^{14} + \frac{1}{5} a^{13} + \frac{1}{5} a^{12} + \frac{2}{5} a^{11} - \frac{2}{5} a^{10} - \frac{1}{5} a^{9} - \frac{2}{5} a^{8} + \frac{1}{5} a^{6} - \frac{1}{5} a^{5} - \frac{1}{5} a^{4} + \frac{2}{5} a^{3} + \frac{1}{5} a^{2} + \frac{2}{5} a - \frac{1}{5}$, $\frac{1}{60285429345455} a^{19} + \frac{2800660520458}{60285429345455} a^{18} - \frac{22766771379307}{60285429345455} a^{17} - \frac{24119658680993}{60285429345455} a^{16} - \frac{13643367570871}{60285429345455} a^{15} + \frac{28943967920863}{60285429345455} a^{14} + \frac{5274515321760}{12057085869091} a^{13} + \frac{21971426141161}{60285429345455} a^{12} + \frac{27827110435346}{60285429345455} a^{11} - \frac{808649152934}{60285429345455} a^{10} + \frac{18891349939104}{60285429345455} a^{9} - \frac{22096813341318}{60285429345455} a^{8} + \frac{27543082326011}{60285429345455} a^{7} + \frac{14872177942748}{60285429345455} a^{6} + \frac{1099254563136}{12057085869091} a^{5} + \frac{13734086941428}{60285429345455} a^{4} + \frac{6143360869054}{60285429345455} a^{3} - \frac{5207044620174}{60285429345455} a^{2} - \frac{23338422925473}{60285429345455} a + \frac{9901889109051}{60285429345455}$

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:  $15$
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:  \( 6572257.07527 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T755:

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 t20n755 are not computed
Character table for t20n755 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 $20$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }$ $20$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ $20$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ $20$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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
$5$5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$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