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Properties

Label 20.12.7382849022...9344.1
Degree $20$
Signature $[12, 4]$
Discriminant $2^{42}\cdot 11^{8}\cdot 23^{8}$
Root discriminant $39.21$
Ramified primes $2, 11, 23$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T230

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-4, -112, 184, 1280, -736, -4768, -328, 4128, -1329, -1060, 2576, 112, -341, 296, -104, -16, -23, -20, 8, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 8*x^18 - 20*x^17 - 23*x^16 - 16*x^15 - 104*x^14 + 296*x^13 - 341*x^12 + 112*x^11 + 2576*x^10 - 1060*x^9 - 1329*x^8 + 4128*x^7 - 328*x^6 - 4768*x^5 - 736*x^4 + 1280*x^3 + 184*x^2 - 112*x - 4)
 
gp: K = bnfinit(x^20 + 8*x^18 - 20*x^17 - 23*x^16 - 16*x^15 - 104*x^14 + 296*x^13 - 341*x^12 + 112*x^11 + 2576*x^10 - 1060*x^9 - 1329*x^8 + 4128*x^7 - 328*x^6 - 4768*x^5 - 736*x^4 + 1280*x^3 + 184*x^2 - 112*x - 4, 1)
 

Normalized defining polynomial

\( x^{20} + 8 x^{18} - 20 x^{17} - 23 x^{16} - 16 x^{15} - 104 x^{14} + 296 x^{13} - 341 x^{12} + 112 x^{11} + 2576 x^{10} - 1060 x^{9} - 1329 x^{8} + 4128 x^{7} - 328 x^{6} - 4768 x^{5} - 736 x^{4} + 1280 x^{3} + 184 x^{2} - 112 x - 4 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(73828490224822538451051656249344=2^{42}\cdot 11^{8}\cdot 23^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $39.21$
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:  $2, 11, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{4} a^{7} - \frac{1}{2} a^{6} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{12} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{4} a^{13} + \frac{1}{4} a^{5} - \frac{1}{2} a$, $\frac{1}{4} a^{14} + \frac{1}{4} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{15} - \frac{1}{4} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{8} a^{16} + \frac{1}{8} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{184} a^{17} + \frac{1}{46} a^{16} - \frac{3}{46} a^{15} - \frac{3}{46} a^{14} - \frac{3}{46} a^{13} - \frac{1}{23} a^{12} + \frac{3}{46} a^{11} - \frac{5}{46} a^{10} - \frac{15}{184} a^{9} + \frac{11}{46} a^{8} + \frac{7}{46} a^{7} - \frac{3}{46} a^{6} - \frac{1}{92} a^{5} - \frac{15}{46} a^{4} + \frac{6}{23} a^{3} - \frac{7}{23} a^{2} + \frac{4}{23} a - \frac{5}{23}$, $\frac{1}{368} a^{18} - \frac{5}{368} a^{16} + \frac{9}{92} a^{15} - \frac{5}{184} a^{14} + \frac{5}{46} a^{13} - \frac{1}{184} a^{12} + \frac{3}{46} a^{11} - \frac{27}{368} a^{10} - \frac{5}{23} a^{9} - \frac{33}{368} a^{8} + \frac{15}{92} a^{7} - \frac{1}{2} a^{6} + \frac{5}{46} a^{5} - \frac{43}{92} a^{4} + \frac{15}{46} a^{3} + \frac{41}{92} a^{2} + \frac{1}{23} a - \frac{29}{92}$, $\frac{1}{367107274536498986752} a^{19} + \frac{422101298898563837}{367107274536498986752} a^{18} - \frac{1088721444808367}{367107274536498986752} a^{17} + \frac{16874764816992473849}{367107274536498986752} a^{16} - \frac{4662762937470176353}{183553637268249493376} a^{15} + \frac{797810880796663933}{7980592924706499712} a^{14} + \frac{2314720093702800603}{183553637268249493376} a^{13} + \frac{10854760422992876195}{183553637268249493376} a^{12} + \frac{36524962546794388633}{367107274536498986752} a^{11} - \frac{28894291304131762459}{367107274536498986752} a^{10} - \frac{59571607720320063007}{367107274536498986752} a^{9} + \frac{29148199116165646457}{367107274536498986752} a^{8} + \frac{2435128858851173785}{91776818634124746688} a^{7} + \frac{25681050018612782685}{91776818634124746688} a^{6} + \frac{10237846460401119015}{91776818634124746688} a^{5} + \frac{35630141009078401283}{91776818634124746688} a^{4} - \frac{29265095288058049217}{91776818634124746688} a^{3} + \frac{35698710912577638211}{91776818634124746688} a^{2} - \frac{1456180849398985893}{2960542536584669248} a - \frac{37549811343652300139}{91776818634124746688}$

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:  \( 8078840883.94 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T230:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 1920
The 24 conjugacy class representatives for t20n230
Character table for t20n230 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 5.5.16386304.1, 10.10.2148087670243328.1, 10.6.8592350680973312.1, 10.6.1074043835121664.1

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

Sibling fields

Degree 10 sibling: data not computed
Degree 20 siblings: data not computed
Degree 30 siblings: data not computed
Degree 32 sibling: 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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$2$2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.8.18.56$x^{8} + 168 x^{4} + 912$$8$$1$$18$$D_4\times C_2$$[2, 2, 3]^{2}$
2.8.18.56$x^{8} + 168 x^{4} + 912$$8$$1$$18$$D_4\times C_2$$[2, 2, 3]^{2}$
$11$11.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
11.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
11.6.4.1$x^{6} + 220 x^{3} + 41503$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
11.6.4.1$x^{6} + 220 x^{3} + 41503$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
$23$23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.3.2.1$x^{3} - 23$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
23.3.2.1$x^{3} - 23$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
23.3.2.1$x^{3} - 23$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
23.3.2.1$x^{3} - 23$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$