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Properties

Label 20.0.93156735146...7728.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{30}\cdot 11^{8}\cdot 337\cdot 1201$
Root discriminant $14.08$
Ramified primes $2, 11, 337, 1201$
Class number $1$
Class group Trivial
Galois group 20T647

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -2, 5, -8, 15, -16, 41, -26, 67, -22, 46, -2, 9, -4, 3, -10, 1, -2, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^17 + x^16 - 10*x^15 + 3*x^14 - 4*x^13 + 9*x^12 - 2*x^11 + 46*x^10 - 22*x^9 + 67*x^8 - 26*x^7 + 41*x^6 - 16*x^5 + 15*x^4 - 8*x^3 + 5*x^2 - 2*x + 1)
 
gp: K = bnfinit(x^20 - 2*x^17 + x^16 - 10*x^15 + 3*x^14 - 4*x^13 + 9*x^12 - 2*x^11 + 46*x^10 - 22*x^9 + 67*x^8 - 26*x^7 + 41*x^6 - 16*x^5 + 15*x^4 - 8*x^3 + 5*x^2 - 2*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{17} + x^{16} - 10 x^{15} + 3 x^{14} - 4 x^{13} + 9 x^{12} - 2 x^{11} + 46 x^{10} - 22 x^{9} + 67 x^{8} - 26 x^{7} + 41 x^{6} - 16 x^{5} + 15 x^{4} - 8 x^{3} + 5 x^{2} - 2 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(93156735146378005577728=2^{30}\cdot 11^{8}\cdot 337\cdot 1201\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $14.08$
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:  $2, 11, 337, 1201$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{81243803} a^{19} + \frac{24840710}{81243803} a^{18} - \frac{28146425}{81243803} a^{17} + \frac{689190}{81243803} a^{16} + \frac{31025332}{81243803} a^{15} - \frac{33181755}{81243803} a^{14} + \frac{140029}{81243803} a^{13} - \frac{33644859}{81243803} a^{12} - \frac{34494217}{81243803} a^{11} + \frac{405480}{81243803} a^{10} - \frac{33117488}{81243803} a^{9} - \frac{2679376}{81243803} a^{8} + \frac{2266206}{81243803} a^{7} + \frac{9972322}{81243803} a^{6} - \frac{27216806}{81243803} a^{5} - \frac{16910842}{81243803} a^{4} + \frac{17206526}{81243803} a^{3} - \frac{25208700}{81243803} a^{2} - \frac{39475728}{81243803} a + \frac{35249439}{81243803}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Trivial group, which has order $1$

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $9$
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 1105.41714709 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T647:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 51200
The 152 conjugacy class representatives for t20n647 are not computed
Character table for t20n647 is not computed

Intermediate fields

\(\Q(\sqrt{-2}) \), 10.0.479756288.1

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 ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R $20$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$2.10.15.9$x^{10} - 6 x^{8} - 24 x^{6} + 80 x^{4} + 336 x^{2} + 33056$$2$$5$$15$$C_{10}$$[3]^{5}$
2.10.15.9$x^{10} - 6 x^{8} - 24 x^{6} + 80 x^{4} + 336 x^{2} + 33056$$2$$5$$15$$C_{10}$$[3]^{5}$
$11$11.10.0.1$x^{10} + x^{2} - x + 6$$1$$10$$0$$C_{10}$$[\ ]^{10}$
11.10.8.3$x^{10} - 11 x^{5} + 847$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
337Data not computed
1201Data not computed