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Properties

Label 20.2.86320850517...1875.1
Degree $20$
Signature $[2, 9]$
Discriminant $-\,5^{14}\cdot 11^{4}\cdot 9931^{5}$
Root discriminant $49.75$
Ramified primes $5, 11, 9931$
Class number $7$ (GRH)
Class group $[7]$ (GRH)
Galois group 20T1023

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![911, -4786, 13439, -41289, 94004, -148291, 182383, -165225, 123896, -70531, 33862, -14279, 5135, -1498, -157, 181, 65, -27, -4, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 - 4*x^18 - 27*x^17 + 65*x^16 + 181*x^15 - 157*x^14 - 1498*x^13 + 5135*x^12 - 14279*x^11 + 33862*x^10 - 70531*x^9 + 123896*x^8 - 165225*x^7 + 182383*x^6 - 148291*x^5 + 94004*x^4 - 41289*x^3 + 13439*x^2 - 4786*x + 911)
 
gp: K = bnfinit(x^20 - x^19 - 4*x^18 - 27*x^17 + 65*x^16 + 181*x^15 - 157*x^14 - 1498*x^13 + 5135*x^12 - 14279*x^11 + 33862*x^10 - 70531*x^9 + 123896*x^8 - 165225*x^7 + 182383*x^6 - 148291*x^5 + 94004*x^4 - 41289*x^3 + 13439*x^2 - 4786*x + 911, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} - 4 x^{18} - 27 x^{17} + 65 x^{16} + 181 x^{15} - 157 x^{14} - 1498 x^{13} + 5135 x^{12} - 14279 x^{11} + 33862 x^{10} - 70531 x^{9} + 123896 x^{8} - 165225 x^{7} + 182383 x^{6} - 148291 x^{5} + 94004 x^{4} - 41289 x^{3} + 13439 x^{2} - 4786 x + 911 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-8632085051759631012385809326171875=-\,5^{14}\cdot 11^{4}\cdot 9931^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $49.75$
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:  $5, 11, 9931$
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}{2346883999078950430248888634928307725943646882457} a^{19} + \frac{22109897719635514125032696341559480620628314480}{2346883999078950430248888634928307725943646882457} a^{18} - \frac{12789586398807922454006367475575080335847800342}{2346883999078950430248888634928307725943646882457} a^{17} - \frac{440714233917496436750667229096521283174932864421}{2346883999078950430248888634928307725943646882457} a^{16} + \frac{328477664918477897567655983583449869016325754962}{2346883999078950430248888634928307725943646882457} a^{15} + \frac{12237826997577190440706993416608398262613508183}{2346883999078950430248888634928307725943646882457} a^{14} - \frac{896023043816668153670842585478320629085562531355}{2346883999078950430248888634928307725943646882457} a^{13} + \frac{719746542973237368398050181317254986637816638652}{2346883999078950430248888634928307725943646882457} a^{12} - \frac{995572878556623543560490649428149362355231509765}{2346883999078950430248888634928307725943646882457} a^{11} + \frac{785050752338674556100288509597566477480196085856}{2346883999078950430248888634928307725943646882457} a^{10} - \frac{150407130194453643666020315617727940382450272241}{2346883999078950430248888634928307725943646882457} a^{9} + \frac{817808646916569125952602045848218626075094321575}{2346883999078950430248888634928307725943646882457} a^{8} - \frac{660213081805722990533650294677918365427715937968}{2346883999078950430248888634928307725943646882457} a^{7} + \frac{377217455033563277612976462601639991572480164565}{2346883999078950430248888634928307725943646882457} a^{6} + \frac{861257348959896705939224328779411988397070250074}{2346883999078950430248888634928307725943646882457} a^{5} + \frac{591967465798446901726368915192871217658829198278}{2346883999078950430248888634928307725943646882457} a^{4} + \frac{825266124602113833779576113363346344153489576981}{2346883999078950430248888634928307725943646882457} a^{3} - \frac{852031838473713888213115095980892698600301130292}{2346883999078950430248888634928307725943646882457} a^{2} + \frac{444795034795235045818520363399868513455089106945}{2346883999078950430248888634928307725943646882457} a + \frac{315027091366472129299920275908084890596036953763}{2346883999078950430248888634928307725943646882457}$

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

Class group and class number

$C_{7}$, which has order $7$ (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:  $10$
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:  \( 76566924.2911 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1023:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7372800
The 324 conjugacy class representatives for t20n1023 are not computed
Character table for t20n1023 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.10.932312193828125.1

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

Sibling fields

Degree 20 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 $20$ $20$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ $20$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ $20$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.12.10.1$x^{12} + 6 x^{11} + 27 x^{10} + 80 x^{9} + 195 x^{8} + 366 x^{7} + 571 x^{6} + 702 x^{5} + 1005 x^{4} + 1140 x^{3} + 357 x^{2} - 138 x + 44$$6$$2$$10$$D_6$$[\ ]_{6}^{2}$
$11$11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.5.0.1$x^{5} + x^{2} - x + 5$$1$$5$$0$$C_5$$[\ ]^{5}$
11.5.0.1$x^{5} + x^{2} - x + 5$$1$$5$$0$$C_5$$[\ ]^{5}$
11.8.4.1$x^{8} + 484 x^{4} - 1331 x^{2} + 58564$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
9931Data not computed