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Properties

Label 20.0.75570724234...3009.2
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 61^{6}\cdot 397^{4}$
Root discriminant $19.68$
Ramified primes $3, 61, 397$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T368

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -2, 9, -16, 52, -83, 164, -183, 268, -336, 317, -320, 286, -242, 174, -112, 73, -39, 18, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 18*x^18 - 39*x^17 + 73*x^16 - 112*x^15 + 174*x^14 - 242*x^13 + 286*x^12 - 320*x^11 + 317*x^10 - 336*x^9 + 268*x^8 - 183*x^7 + 164*x^6 - 83*x^5 + 52*x^4 - 16*x^3 + 9*x^2 - 2*x + 1)
 
gp: K = bnfinit(x^20 - 5*x^19 + 18*x^18 - 39*x^17 + 73*x^16 - 112*x^15 + 174*x^14 - 242*x^13 + 286*x^12 - 320*x^11 + 317*x^10 - 336*x^9 + 268*x^8 - 183*x^7 + 164*x^6 - 83*x^5 + 52*x^4 - 16*x^3 + 9*x^2 - 2*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} + 18 x^{18} - 39 x^{17} + 73 x^{16} - 112 x^{15} + 174 x^{14} - 242 x^{13} + 286 x^{12} - 320 x^{11} + 317 x^{10} - 336 x^{9} + 268 x^{8} - 183 x^{7} + 164 x^{6} - 83 x^{5} + 52 x^{4} - 16 x^{3} + 9 x^{2} - 2 x + 1 \)

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:  \(75570724234611059753853009=3^{10}\cdot 61^{6}\cdot 397^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $19.68$
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:  $3, 61, 397$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{3690219511145} a^{19} - \frac{884082209249}{3690219511145} a^{18} + \frac{414967285444}{3690219511145} a^{17} + \frac{35552024500}{738043902229} a^{16} + \frac{498567645083}{3690219511145} a^{15} - \frac{289861101554}{3690219511145} a^{14} - \frac{242352352624}{738043902229} a^{13} + \frac{1270696514203}{3690219511145} a^{12} + \frac{1408937117529}{3690219511145} a^{11} + \frac{1552993303259}{3690219511145} a^{10} + \frac{9074610221}{3690219511145} a^{9} - \frac{352257915007}{738043902229} a^{8} + \frac{15999507978}{3690219511145} a^{7} - \frac{213746002411}{738043902229} a^{6} + \frac{1659527481459}{3690219511145} a^{5} + \frac{1543629861836}{3690219511145} a^{4} - \frac{94967045562}{3690219511145} a^{3} - \frac{608132659578}{3690219511145} a^{2} - \frac{1769591929704}{3690219511145} a + \frac{1428971656029}{3690219511145}$

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:  $9$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{1394686377077}{3690219511145} a^{19} + \frac{6509228914628}{3690219511145} a^{18} - \frac{22573695769528}{3690219511145} a^{17} + \frac{8994392037037}{738043902229} a^{16} - \frac{79914032782326}{3690219511145} a^{15} + \frac{114199074712578}{3690219511145} a^{14} - \frac{35214031659632}{738043902229} a^{13} + \frac{235282652929924}{3690219511145} a^{12} - \frac{254530412991608}{3690219511145} a^{11} + \frac{270477554896917}{3690219511145} a^{10} - \frac{245642293088672}{3690219511145} a^{9} + \frac{54436618934112}{738043902229} a^{8} - \frac{171673894838431}{3690219511145} a^{7} + \frac{16635534700580}{738043902229} a^{6} - \frac{108713644780598}{3690219511145} a^{5} + \frac{22935607117768}{3690219511145} a^{4} - \frac{16085225437836}{3690219511145} a^{3} - \frac{13252360163484}{3690219511145} a^{2} - \frac{2383196858422}{3690219511145} a + \frac{265686700087}{3690219511145} \) (order $6$)
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:  \( 90268.3287749 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T368:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7680
The 72 conjugacy class representatives for t20n368 are not computed
Character table for t20n368 is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 5.5.24217.1, 10.0.142510530627.1, 10.4.8693142368247.1, 10.6.35774248429.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$3$3.10.5.2$x^{10} - 81 x^{2} + 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
3.10.5.2$x^{10} - 81 x^{2} + 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$61$61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
61.4.3.2$x^{4} - 244$$4$$1$$3$$C_4$$[\ ]_{4}$
61.4.3.2$x^{4} - 244$$4$$1$$3$$C_4$$[\ ]_{4}$
61.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
397Data not computed