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Properties

Label 20.18.1121061380...5488.1
Degree $20$
Signature $[18, 1]$
Discriminant $-\,2^{20}\cdot 3^{10}\cdot 23^{4}\cdot 421\cdot 1451^{4}\cdot 3467$
Root discriminant $56.56$
Ramified primes $2, 3, 23, 421, 1451, 3467$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1036

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![81, -288, -464, 2492, 333, -8524, 2604, 14846, -7190, -14016, 8643, 7148, -5685, -1674, 2076, -12, -381, 78, 23, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 23*x^18 + 78*x^17 - 381*x^16 - 12*x^15 + 2076*x^14 - 1674*x^13 - 5685*x^12 + 7148*x^11 + 8643*x^10 - 14016*x^9 - 7190*x^8 + 14846*x^7 + 2604*x^6 - 8524*x^5 + 333*x^4 + 2492*x^3 - 464*x^2 - 288*x + 81)
 
gp: K = bnfinit(x^20 - 10*x^19 + 23*x^18 + 78*x^17 - 381*x^16 - 12*x^15 + 2076*x^14 - 1674*x^13 - 5685*x^12 + 7148*x^11 + 8643*x^10 - 14016*x^9 - 7190*x^8 + 14846*x^7 + 2604*x^6 - 8524*x^5 + 333*x^4 + 2492*x^3 - 464*x^2 - 288*x + 81, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 23 x^{18} + 78 x^{17} - 381 x^{16} - 12 x^{15} + 2076 x^{14} - 1674 x^{13} - 5685 x^{12} + 7148 x^{11} + 8643 x^{10} - 14016 x^{9} - 7190 x^{8} + 14846 x^{7} + 2604 x^{6} - 8524 x^{5} + 333 x^{4} + 2492 x^{3} - 464 x^{2} - 288 x + 81 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[18, 1]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-112106138048743837664176950752575488=-\,2^{20}\cdot 3^{10}\cdot 23^{4}\cdot 421\cdot 1451^{4}\cdot 3467\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $56.56$
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:  $2, 3, 23, 421, 1451, 3467$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{189} a^{18} - \frac{1}{21} a^{17} - \frac{22}{189} a^{16} + \frac{2}{189} a^{15} + \frac{5}{27} a^{14} - \frac{7}{27} a^{13} + \frac{11}{189} a^{12} - \frac{88}{189} a^{11} + \frac{68}{189} a^{10} - \frac{11}{189} a^{9} - \frac{53}{189} a^{8} - \frac{65}{189} a^{7} - \frac{55}{189} a^{6} - \frac{68}{189} a^{5} - \frac{20}{189} a^{4} - \frac{16}{63} a^{3} + \frac{20}{63} a^{2} - \frac{67}{189} a - \frac{5}{21}$, $\frac{1}{189} a^{19} + \frac{86}{189} a^{17} - \frac{1}{27} a^{16} + \frac{53}{189} a^{15} + \frac{11}{27} a^{14} - \frac{52}{189} a^{13} + \frac{11}{189} a^{12} + \frac{32}{189} a^{11} + \frac{34}{189} a^{10} + \frac{37}{189} a^{9} + \frac{25}{189} a^{8} - \frac{73}{189} a^{7} + \frac{4}{189} a^{6} - \frac{65}{189} a^{5} - \frac{13}{63} a^{4} + \frac{2}{63} a^{3} - \frac{94}{189} a^{2} - \frac{3}{7} a - \frac{1}{7}$

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:  $18$
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:  \( 245620475404 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1036:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 14745600
The 396 conjugacy class representatives for t20n1036 are not computed
Character table for t20n1036 is not computed

Intermediate fields

\(\Q(\sqrt{3}) \), 10.10.277138413923328.1

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 R R ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }$ ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/13.10.0.1}{10} }{,}\,{\href{/LocalNumberField/13.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ R $20$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ $20$ $20$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
$3$3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$23$23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.3.0.1$x^{3} - x + 4$$1$$3$$0$$C_3$$[\ ]^{3}$
23.3.0.1$x^{3} - x + 4$$1$$3$$0$$C_3$$[\ ]^{3}$
23.6.0.1$x^{6} - x + 15$$1$$6$$0$$C_6$$[\ ]^{6}$
421Data not computed
1451Data not computed
3467Data not computed