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Properties

Label 20.12.6077381409...0625.1
Degree $20$
Signature $[12, 4]$
Discriminant $5^{17}\cdot 6029^{5}$
Root discriminant $34.61$
Ramified primes $5, 6029$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T797

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 15, -80, 25, 640, -1553, 255, 4745, -8790, 5125, 4014, -8595, 5170, -225, -1355, 788, -175, -30, 35, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 35*x^18 - 30*x^17 - 175*x^16 + 788*x^15 - 1355*x^14 - 225*x^13 + 5170*x^12 - 8595*x^11 + 4014*x^10 + 5125*x^9 - 8790*x^8 + 4745*x^7 + 255*x^6 - 1553*x^5 + 640*x^4 + 25*x^3 - 80*x^2 + 15*x + 1)
 
gp: K = bnfinit(x^20 - 10*x^19 + 35*x^18 - 30*x^17 - 175*x^16 + 788*x^15 - 1355*x^14 - 225*x^13 + 5170*x^12 - 8595*x^11 + 4014*x^10 + 5125*x^9 - 8790*x^8 + 4745*x^7 + 255*x^6 - 1553*x^5 + 640*x^4 + 25*x^3 - 80*x^2 + 15*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 35 x^{18} - 30 x^{17} - 175 x^{16} + 788 x^{15} - 1355 x^{14} - 225 x^{13} + 5170 x^{12} - 8595 x^{11} + 4014 x^{10} + 5125 x^{9} - 8790 x^{8} + 4745 x^{7} + 255 x^{6} - 1553 x^{5} + 640 x^{4} + 25 x^{3} - 80 x^{2} + 15 x + 1 \)

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:  \(6077381409667160910797119140625=5^{17}\cdot 6029^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $34.61$
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:  $5, 6029$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} - \frac{1}{5} a^{11} - \frac{1}{5} a^{10} - \frac{1}{5} a^{7} + \frac{1}{5} a^{6} + \frac{1}{5} a^{5} - \frac{1}{5} a^{2} + \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{13} - \frac{2}{5} a^{11} - \frac{1}{5} a^{10} - \frac{1}{5} a^{8} + \frac{2}{5} a^{6} + \frac{1}{5} a^{5} - \frac{1}{5} a^{3} + \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{14} + \frac{2}{5} a^{11} - \frac{2}{5} a^{10} - \frac{1}{5} a^{9} - \frac{2}{5} a^{6} + \frac{2}{5} a^{5} - \frac{1}{5} a^{4} - \frac{2}{5} a + \frac{2}{5}$, $\frac{1}{5} a^{15} + \frac{1}{5} a^{10} + \frac{2}{5} a^{5} - \frac{2}{5}$, $\frac{1}{25} a^{16} + \frac{2}{25} a^{15} - \frac{4}{25} a^{11} - \frac{3}{25} a^{10} + \frac{1}{5} a^{8} + \frac{1}{5} a^{7} - \frac{8}{25} a^{6} - \frac{1}{25} a^{5} + \frac{1}{5} a^{4} + \frac{1}{5} a^{2} - \frac{7}{25} a + \frac{6}{25}$, $\frac{1}{25} a^{17} + \frac{1}{25} a^{15} + \frac{1}{25} a^{12} + \frac{6}{25} a^{10} + \frac{1}{5} a^{9} - \frac{1}{5} a^{8} + \frac{2}{25} a^{7} - \frac{1}{5} a^{6} - \frac{3}{25} a^{5} - \frac{2}{5} a^{4} + \frac{1}{5} a^{3} + \frac{3}{25} a^{2} + \frac{8}{25}$, $\frac{1}{12266425} a^{18} - \frac{9}{12266425} a^{17} + \frac{149228}{12266425} a^{16} - \frac{6452}{66305} a^{15} - \frac{222217}{2453285} a^{14} - \frac{774189}{12266425} a^{13} - \frac{1052409}{12266425} a^{12} - \frac{6035442}{12266425} a^{11} + \frac{461678}{2453285} a^{10} - \frac{751443}{2453285} a^{9} - \frac{4520228}{12266425} a^{8} + \frac{4287962}{12266425} a^{7} - \frac{2465959}{12266425} a^{6} - \frac{2819}{490657} a^{5} - \frac{994172}{2453285} a^{4} + \frac{3627973}{12266425} a^{3} + \frac{3776383}{12266425} a^{2} - \frac{5371441}{12266425} a + \frac{474407}{2453285}$, $\frac{1}{12266425} a^{19} + \frac{4031}{331525} a^{17} + \frac{149432}{12266425} a^{16} + \frac{82552}{2453285} a^{15} - \frac{960814}{12266425} a^{14} - \frac{132051}{2453285} a^{13} - \frac{787413}{12266425} a^{12} - \frac{491603}{12266425} a^{11} - \frac{6188}{490657} a^{10} + \frac{917397}{12266425} a^{9} + \frac{1062351}{2453285} a^{8} - \frac{3126861}{12266425} a^{7} - \frac{184541}{12266425} a^{6} - \frac{139713}{2453285} a^{5} - \frac{1857207}{12266425} a^{4} + \frac{907087}{2453285} a^{3} + \frac{1629871}{12266425} a^{2} + \frac{641481}{12266425} a + \frac{344407}{2453285}$

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:  \( 149292590.811 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T797:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 122880
The 108 conjugacy class representatives for t20n797 are not computed
Character table for t20n797 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.753625.1, 10.10.2839753203125.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$5$5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.12.11.2$x^{12} - 20$$12$$1$$11$$S_3 \times C_4$$[\ ]_{12}^{2}$
6029Data not computed