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Properties

Label 20.4.11145552632...4736.1
Degree $20$
Signature $[4, 8]$
Discriminant $2^{36}\cdot 7^{6}\cdot 13^{10}$
Root discriminant $22.51$
Ramified primes $2, 7, 13$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T199

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-4, -16, -12, -8, -122, -316, -410, -296, 81, 230, -103, -180, 46, 94, 13, -44, -6, 22, -3, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 - 3*x^18 + 22*x^17 - 6*x^16 - 44*x^15 + 13*x^14 + 94*x^13 + 46*x^12 - 180*x^11 - 103*x^10 + 230*x^9 + 81*x^8 - 296*x^7 - 410*x^6 - 316*x^5 - 122*x^4 - 8*x^3 - 12*x^2 - 16*x - 4)
 
gp: K = bnfinit(x^20 - 4*x^19 - 3*x^18 + 22*x^17 - 6*x^16 - 44*x^15 + 13*x^14 + 94*x^13 + 46*x^12 - 180*x^11 - 103*x^10 + 230*x^9 + 81*x^8 - 296*x^7 - 410*x^6 - 316*x^5 - 122*x^4 - 8*x^3 - 12*x^2 - 16*x - 4, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} - 3 x^{18} + 22 x^{17} - 6 x^{16} - 44 x^{15} + 13 x^{14} + 94 x^{13} + 46 x^{12} - 180 x^{11} - 103 x^{10} + 230 x^{9} + 81 x^{8} - 296 x^{7} - 410 x^{6} - 316 x^{5} - 122 x^{4} - 8 x^{3} - 12 x^{2} - 16 x - 4 \)

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:  \(1114555263208692764939124736=2^{36}\cdot 7^{6}\cdot 13^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $22.51$
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:  $2, 7, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{14} - \frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{15} - \frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{14} - \frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{1630957845947002} a^{19} + \frac{72622575670779}{1630957845947002} a^{18} + \frac{146035327937286}{815478922973501} a^{17} + \frac{116337080655375}{815478922973501} a^{16} + \frac{449636066518113}{1630957845947002} a^{15} + \frac{199990310026783}{1630957845947002} a^{14} - \frac{202348592910573}{1630957845947002} a^{13} - \frac{562023790661449}{1630957845947002} a^{12} + \frac{346753378292071}{1630957845947002} a^{11} - \frac{492462709693653}{1630957845947002} a^{10} + \frac{629002986243201}{1630957845947002} a^{9} + \frac{119379439375783}{1630957845947002} a^{8} - \frac{152225375823380}{815478922973501} a^{7} - \frac{335903983730142}{815478922973501} a^{6} + \frac{432730198305627}{1630957845947002} a^{5} + \frac{634515360558601}{1630957845947002} a^{4} - \frac{35897660522374}{815478922973501} a^{3} + \frac{371163433571669}{815478922973501} a^{2} - \frac{163125981535558}{815478922973501} a + \frac{386547842597013}{815478922973501}$

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:  \( 582590.858531 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T199:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 1440
The 22 conjugacy class representatives for t20n199
Character table for t20n199 is not computed

Intermediate fields

\(\Q(\sqrt{13}) \), 10.2.197544116224.1

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

Sibling fields

Degree 12 siblings: data not computed
Degree 20 siblings: data not computed
Degree 24 siblings: data not computed
Degree 30 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 ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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
$2$2.8.16.9$x^{8} + 2 x^{4} + 8 x + 12$$4$$2$$16$$S_4$$[8/3, 8/3]_{3}^{2}$
2.12.20.52$x^{12} - 5 x^{8} + 4 x^{6} + 3 x^{4} + 8 x^{2} - 7$$6$$2$$20$$C_2\times S_4$$[2, 8/3, 8/3]_{3}^{2}$
$7$7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.6.0.1$x^{6} + 3 x^{2} - x + 5$$1$$6$$0$$C_6$$[\ ]^{6}$
7.12.6.1$x^{12} + 294 x^{8} + 3430 x^{6} + 21609 x^{4} + 487403 x^{2} + 2941225$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$13$13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.6.3.1$x^{6} - 52 x^{4} + 676 x^{2} - 79092$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
13.12.6.1$x^{12} + 338 x^{8} + 8788 x^{6} + 28561 x^{4} + 19307236$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$