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Properties

Label 20.8.58762130557...0625.1
Degree $20$
Signature $[8, 6]$
Discriminant $3^{2}\cdot 5^{10}\cdot 401^{8}$
Root discriminant $27.44$
Ramified primes $3, 5, 401$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T141

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 1, -1, 0, -14, -3, -29, 22, 4, 78, 87, -78, 4, -22, -29, 3, -14, 0, -1, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 - x^18 - 14*x^16 + 3*x^15 - 29*x^14 - 22*x^13 + 4*x^12 - 78*x^11 + 87*x^10 + 78*x^9 + 4*x^8 + 22*x^7 - 29*x^6 - 3*x^5 - 14*x^4 - x^2 + x + 1)
 
gp: K = bnfinit(x^20 - x^19 - x^18 - 14*x^16 + 3*x^15 - 29*x^14 - 22*x^13 + 4*x^12 - 78*x^11 + 87*x^10 + 78*x^9 + 4*x^8 + 22*x^7 - 29*x^6 - 3*x^5 - 14*x^4 - x^2 + x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} - x^{18} - 14 x^{16} + 3 x^{15} - 29 x^{14} - 22 x^{13} + 4 x^{12} - 78 x^{11} + 87 x^{10} + 78 x^{9} + 4 x^{8} + 22 x^{7} - 29 x^{6} - 3 x^{5} - 14 x^{4} - x^{2} + x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(58762130557815394031337890625=3^{2}\cdot 5^{10}\cdot 401^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.44$
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:  $3, 5, 401$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{3} a^{16} - \frac{1}{3} a^{15} - \frac{1}{3} a^{14} - \frac{1}{3} a^{12} - \frac{1}{3} a^{11} - \frac{1}{3} a^{9} - \frac{1}{3} a^{7} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{17} + \frac{1}{3} a^{15} - \frac{1}{3} a^{14} - \frac{1}{3} a^{13} + \frac{1}{3} a^{12} - \frac{1}{3} a^{11} - \frac{1}{3} a^{10} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{41387781} a^{18} + \frac{1362551}{41387781} a^{17} + \frac{1101908}{41387781} a^{16} - \frac{807881}{13795927} a^{15} + \frac{20122802}{41387781} a^{14} + \frac{19060943}{41387781} a^{13} + \frac{3319844}{13795927} a^{12} - \frac{19470673}{41387781} a^{11} + \frac{2557933}{13795927} a^{10} + \frac{11426924}{41387781} a^{9} - \frac{2557933}{13795927} a^{8} - \frac{19470673}{41387781} a^{7} + \frac{3836395}{41387781} a^{6} - \frac{2843637}{13795927} a^{5} - \frac{20122802}{41387781} a^{4} - \frac{16219570}{41387781} a^{3} + \frac{12694019}{41387781} a^{2} + \frac{5052826}{13795927} a + \frac{4598642}{13795927}$, $\frac{1}{538041153} a^{19} - \frac{2}{179347051} a^{18} + \frac{20779099}{538041153} a^{17} - \frac{4146989}{538041153} a^{16} - \frac{78368859}{179347051} a^{15} + \frac{62513474}{538041153} a^{14} + \frac{214532525}{538041153} a^{13} + \frac{26543559}{179347051} a^{12} - \frac{74596449}{179347051} a^{11} - \frac{57839746}{538041153} a^{10} - \frac{25684196}{179347051} a^{9} + \frac{91183778}{538041153} a^{8} - \frac{269014532}{538041153} a^{7} + \frac{168286279}{538041153} a^{6} - \frac{119060203}{538041153} a^{5} - \frac{225223225}{538041153} a^{4} + \frac{60975034}{538041153} a^{3} - \frac{162465227}{538041153} a^{2} - \frac{187261715}{538041153} a - \frac{219372275}{538041153}$

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:  \( 8057608.88809 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T141:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 640
The 40 conjugacy class representatives for t20n141
Character table for t20n141 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.160801.1, 10.10.80803005003125.1, 10.4.242409015009375.1, 10.4.77570884803.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ R R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$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}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5Data not computed
401Data not computed