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Properties

Label 20.6.78574805586...9483.6
Degree $20$
Signature $[6, 7]$
Discriminant $-\,13^{10}\cdot 97^{2}\cdot 347^{7}$
Root discriminant $44.13$
Ramified primes $13, 97, 347$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T796

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1237531, -1792952, 1358059, 1292806, -463938, 206790, 243060, -141925, -31665, -22238, -18629, 11695, -165, -188, 1395, -119, -197, 14, -1, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 - x^18 + 14*x^17 - 197*x^16 - 119*x^15 + 1395*x^14 - 188*x^13 - 165*x^12 + 11695*x^11 - 18629*x^10 - 22238*x^9 - 31665*x^8 - 141925*x^7 + 243060*x^6 + 206790*x^5 - 463938*x^4 + 1292806*x^3 + 1358059*x^2 - 1792952*x - 1237531)
 
gp: K = bnfinit(x^20 - x^19 - x^18 + 14*x^17 - 197*x^16 - 119*x^15 + 1395*x^14 - 188*x^13 - 165*x^12 + 11695*x^11 - 18629*x^10 - 22238*x^9 - 31665*x^8 - 141925*x^7 + 243060*x^6 + 206790*x^5 - 463938*x^4 + 1292806*x^3 + 1358059*x^2 - 1792952*x - 1237531, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} - x^{18} + 14 x^{17} - 197 x^{16} - 119 x^{15} + 1395 x^{14} - 188 x^{13} - 165 x^{12} + 11695 x^{11} - 18629 x^{10} - 22238 x^{9} - 31665 x^{8} - 141925 x^{7} + 243060 x^{6} + 206790 x^{5} - 463938 x^{4} + 1292806 x^{3} + 1358059 x^{2} - 1792952 x - 1237531 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-785748055861331850537130300409483=-\,13^{10}\cdot 97^{2}\cdot 347^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $44.13$
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:  $13, 97, 347$
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}{3406110016729151575120228138948508154168705548283847128985894873} a^{19} + \frac{574090342078123476896945863765802399093166619431828662487577935}{3406110016729151575120228138948508154168705548283847128985894873} a^{18} - \frac{13402890047643248545545286444917767344024496104921790061183195}{55837869126707402870823412113909969740470582758751592278457293} a^{17} + \frac{1263416509631067903339576470721953747244982677443177778989413906}{3406110016729151575120228138948508154168705548283847128985894873} a^{16} - \frac{1223739053582628320243184581146610950344627481501283709171370171}{3406110016729151575120228138948508154168705548283847128985894873} a^{15} + \frac{1478457220856001987899942295001098494707258076211699243865754544}{3406110016729151575120228138948508154168705548283847128985894873} a^{14} - \frac{929274598107629712105631278114237367274620960971645233285339112}{3406110016729151575120228138948508154168705548283847128985894873} a^{13} + \frac{1225938782533786052708357472982981115259559004447953755595114225}{3406110016729151575120228138948508154168705548283847128985894873} a^{12} + \frac{235208201035698198257347814649639364774596926389398585770131357}{3406110016729151575120228138948508154168705548283847128985894873} a^{11} + \frac{351310357341282401060892017735947424061272727196398786068508051}{3406110016729151575120228138948508154168705548283847128985894873} a^{10} - \frac{1215348249645174662319532421938989277974085288550142900409963792}{3406110016729151575120228138948508154168705548283847128985894873} a^{9} + \frac{357572243515374056355707253458109030703602131693281356868971497}{3406110016729151575120228138948508154168705548283847128985894873} a^{8} + \frac{1601051128923824432316619071878219672702809399906522846738868010}{3406110016729151575120228138948508154168705548283847128985894873} a^{7} - \frac{95468824283583659586018076478875115992998356244750448650085701}{3406110016729151575120228138948508154168705548283847128985894873} a^{6} - \frac{1204943194357984930917867790350907190817436635170199481028522076}{3406110016729151575120228138948508154168705548283847128985894873} a^{5} - \frac{291452581242951546362882764147590395887761420226704420492235201}{3406110016729151575120228138948508154168705548283847128985894873} a^{4} + \frac{598939879116086123233887755955129964359611014609189136457738982}{3406110016729151575120228138948508154168705548283847128985894873} a^{3} - \frac{690977358010688362388107222689227593063111145302649294102485895}{3406110016729151575120228138948508154168705548283847128985894873} a^{2} - \frac{1334944480204429497150243310685813876746123658184122244208000198}{3406110016729151575120228138948508154168705548283847128985894873} a + \frac{1208001695695981529213791606333237391779744164673614721012752591}{3406110016729151575120228138948508154168705548283847128985894873}$

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:  $12$
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:  \( 402282811.058 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T796:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 122880
The 108 conjugacy class representatives for t20n796 are not computed
Character table for t20n796 is not computed

Intermediate fields

\(\Q(\sqrt{13}) \), 5.3.4511.1, 10.6.44707018837.1

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 $20$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ $20$ $20$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$13$13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$97$97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.4.2.2$x^{4} - 97 x^{2} + 47045$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
347Data not computed