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Properties

Label 20.12.1522267764...0000.1
Degree $20$
Signature $[12, 4]$
Discriminant $2^{16}\cdot 5^{22}\cdot 7^{8}\cdot 13^{2}$
Root discriminant $28.78$
Ramified primes $2, 5, 7, 13$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T140

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 130, 185, -340, -345, 126, 530, -70, -465, 460, -355, -100, 645, -470, -120, 308, -115, -30, 35, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 35*x^18 - 30*x^17 - 115*x^16 + 308*x^15 - 120*x^14 - 470*x^13 + 645*x^12 - 100*x^11 - 355*x^10 + 460*x^9 - 465*x^8 - 70*x^7 + 530*x^6 + 126*x^5 - 345*x^4 - 340*x^3 + 185*x^2 + 130*x - 1)
 
gp: K = bnfinit(x^20 - 10*x^19 + 35*x^18 - 30*x^17 - 115*x^16 + 308*x^15 - 120*x^14 - 470*x^13 + 645*x^12 - 100*x^11 - 355*x^10 + 460*x^9 - 465*x^8 - 70*x^7 + 530*x^6 + 126*x^5 - 345*x^4 - 340*x^3 + 185*x^2 + 130*x - 1, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 35 x^{18} - 30 x^{17} - 115 x^{16} + 308 x^{15} - 120 x^{14} - 470 x^{13} + 645 x^{12} - 100 x^{11} - 355 x^{10} + 460 x^{9} - 465 x^{8} - 70 x^{7} + 530 x^{6} + 126 x^{5} - 345 x^{4} - 340 x^{3} + 185 x^{2} + 130 x - 1 \)

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:  \(152226776406250000000000000000=2^{16}\cdot 5^{22}\cdot 7^{8}\cdot 13^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $28.78$
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:  $2, 5, 7, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{14} a^{15} + \frac{3}{14} a^{14} + \frac{1}{7} a^{11} + \frac{3}{14} a^{10} - \frac{1}{2} a^{9} - \frac{3}{7} a^{8} - \frac{2}{7} a^{7} - \frac{1}{2} a^{6} - \frac{5}{14} a^{5} - \frac{3}{7} a^{4} - \frac{1}{2} a^{3} - \frac{3}{7} a^{2} - \frac{1}{2} a + \frac{1}{7}$, $\frac{1}{364} a^{16} - \frac{2}{91} a^{15} + \frac{29}{182} a^{14} - \frac{3}{13} a^{13} + \frac{43}{182} a^{12} + \frac{15}{182} a^{11} + \frac{29}{182} a^{10} + \frac{44}{91} a^{9} + \frac{111}{364} a^{8} - \frac{1}{14} a^{7} - \frac{17}{91} a^{6} - \frac{3}{26} a^{5} - \frac{51}{182} a^{4} - \frac{19}{91} a^{3} + \frac{1}{7} a^{2} - \frac{83}{182} a - \frac{113}{364}$, $\frac{1}{364} a^{17} - \frac{3}{182} a^{15} + \frac{4}{91} a^{14} - \frac{10}{91} a^{13} - \frac{5}{182} a^{12} - \frac{33}{182} a^{11} - \frac{22}{91} a^{10} + \frac{9}{52} a^{9} - \frac{12}{91} a^{8} + \frac{22}{91} a^{7} + \frac{71}{182} a^{6} - \frac{37}{182} a^{5} + \frac{9}{182} a^{4} + \frac{43}{91} a^{3} - \frac{57}{182} a^{2} - \frac{167}{364} a + \frac{3}{182}$, $\frac{1}{364} a^{18} - \frac{3}{182} a^{15} + \frac{11}{182} a^{14} + \frac{8}{91} a^{13} + \frac{43}{182} a^{12} - \frac{19}{182} a^{11} - \frac{57}{364} a^{10} - \frac{3}{13} a^{9} + \frac{1}{7} a^{8} + \frac{16}{91} a^{7} - \frac{59}{182} a^{6} - \frac{1}{2} a^{5} - \frac{25}{182} a^{4} + \frac{79}{182} a^{3} + \frac{171}{364} a^{2} - \frac{20}{91} a - \frac{20}{91}$, $\frac{1}{400764} a^{19} + \frac{541}{400764} a^{18} - \frac{35}{57252} a^{17} + \frac{199}{200382} a^{16} + \frac{1}{4771} a^{15} - \frac{28451}{200382} a^{14} - \frac{12923}{100191} a^{13} + \frac{376}{33397} a^{12} - \frac{4161}{19084} a^{11} - \frac{11053}{57252} a^{10} + \frac{53597}{133588} a^{9} + \frac{17599}{100191} a^{8} - \frac{81173}{200382} a^{7} - \frac{302}{2569} a^{6} - \frac{26125}{100191} a^{5} + \frac{27887}{200382} a^{4} + \frac{5591}{30828} a^{3} - \frac{4331}{133588} a^{2} + \frac{121955}{400764} a - \frac{490}{1101}$

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:  \( 38972536.2748 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T140:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 640
The 22 conjugacy class representatives for t20n140
Character table for t20n140 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.2450000.1, 10.6.390162500000000.1, 10.6.78032500000000.1, 10.10.30012500000000.1

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

Sibling fields

Degree 10 siblings: data not computed
Degree 20 siblings: data not computed
Degree 32 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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ R R ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$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}$
5Data not computed
$7$7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
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}$
7.4.2.2$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}$
$13$13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$