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Properties

Label 20.0.14512989636...4528.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{24}\cdot 13^{17}$
Root discriminant $20.33$
Ramified primes $2, 13$
Class number $2$
Class group $[2]$
Galois group $C_2^4:C_5:C_4$ (as 20T88)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -16, 373, -1730, 4525, -8494, 12190, -13450, 11101, -6002, 765, 2262, -2581, 1392, -202, -320, 317, -160, 51, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 51*x^18 - 160*x^17 + 317*x^16 - 320*x^15 - 202*x^14 + 1392*x^13 - 2581*x^12 + 2262*x^11 + 765*x^10 - 6002*x^9 + 11101*x^8 - 13450*x^7 + 12190*x^6 - 8494*x^5 + 4525*x^4 - 1730*x^3 + 373*x^2 - 16*x + 1)
 
gp: K = bnfinit(x^20 - 10*x^19 + 51*x^18 - 160*x^17 + 317*x^16 - 320*x^15 - 202*x^14 + 1392*x^13 - 2581*x^12 + 2262*x^11 + 765*x^10 - 6002*x^9 + 11101*x^8 - 13450*x^7 + 12190*x^6 - 8494*x^5 + 4525*x^4 - 1730*x^3 + 373*x^2 - 16*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 51 x^{18} - 160 x^{17} + 317 x^{16} - 320 x^{15} - 202 x^{14} + 1392 x^{13} - 2581 x^{12} + 2262 x^{11} + 765 x^{10} - 6002 x^{9} + 11101 x^{8} - 13450 x^{7} + 12190 x^{6} - 8494 x^{5} + 4525 x^{4} - 1730 x^{3} + 373 x^{2} - 16 x + 1 \)

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:  \(145129896369299292870934528=2^{24}\cdot 13^{17}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.33$
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:  $2, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{200829049725681589126097} a^{19} + \frac{54620935363624334229219}{200829049725681589126097} a^{18} - \frac{2347796561082290801415}{200829049725681589126097} a^{17} - \frac{47440268792672193468639}{200829049725681589126097} a^{16} - \frac{65099906132299389517886}{200829049725681589126097} a^{15} - \frac{65094357035432420629425}{200829049725681589126097} a^{14} - \frac{16513309344056874458770}{200829049725681589126097} a^{13} - \frac{47031249490101067268447}{200829049725681589126097} a^{12} - \frac{14924399510489952123114}{200829049725681589126097} a^{11} + \frac{95205341577600154726645}{200829049725681589126097} a^{10} + \frac{97933717703428691914665}{200829049725681589126097} a^{9} + \frac{55311436065115085234825}{200829049725681589126097} a^{8} - \frac{9364690324582282713110}{200829049725681589126097} a^{7} + \frac{9393073709359773552470}{200829049725681589126097} a^{6} + \frac{25160657440029754870679}{200829049725681589126097} a^{5} - \frac{40501630595976862009478}{200829049725681589126097} a^{4} + \frac{80918371246416850552251}{200829049725681589126097} a^{3} + \frac{18852032215204689369190}{200829049725681589126097} a^{2} - \frac{28098990035194215720858}{200829049725681589126097} a + \frac{45182713170674342992959}{200829049725681589126097}$

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

Class group and class number

$C_{2}$, which has order $2$

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:  \( 42936.0339913 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4:C_5:C_4$ (as 20T88):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 320
The 11 conjugacy class representatives for $C_2^4:C_5:C_4$
Character table for $C_2^4:C_5:C_4$

Intermediate fields

\(\Q(\sqrt{13}) \), 5.1.35152.1, 10.2.16063620352.1

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

Sibling fields

Degree 10 siblings: data not computed
Degree 16 sibling: data not computed
Degree 20 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 ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{5}$

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
2Data not computed
$13$13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
13.8.7.2$x^{8} - 52$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
13.8.7.2$x^{8} - 52$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$