↑

Properties

Label 20.8.24403618437...8125.1
Degree $20$
Signature $[8, 6]$
Discriminant $5^{15}\cdot 97^{2}\cdot 419^{2}\cdot 695771^{2}$
Root discriminant $37.10$
Ramified primes $5, 97, 419, 695771$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1039

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-5429, -11078, -9048, 2464, 21334, 29732, 18000, -231, -14173, -12637, -3171, 883, 1667, 29, -760, 115, 216, -6, -26, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 - 26*x^18 - 6*x^17 + 216*x^16 + 115*x^15 - 760*x^14 + 29*x^13 + 1667*x^12 + 883*x^11 - 3171*x^10 - 12637*x^9 - 14173*x^8 - 231*x^7 + 18000*x^6 + 29732*x^5 + 21334*x^4 + 2464*x^3 - 9048*x^2 - 11078*x - 5429)
 
gp: K = bnfinit(x^20 - x^19 - 26*x^18 - 6*x^17 + 216*x^16 + 115*x^15 - 760*x^14 + 29*x^13 + 1667*x^12 + 883*x^11 - 3171*x^10 - 12637*x^9 - 14173*x^8 - 231*x^7 + 18000*x^6 + 29732*x^5 + 21334*x^4 + 2464*x^3 - 9048*x^2 - 11078*x - 5429, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} - 26 x^{18} - 6 x^{17} + 216 x^{16} + 115 x^{15} - 760 x^{14} + 29 x^{13} + 1667 x^{12} + 883 x^{11} - 3171 x^{10} - 12637 x^{9} - 14173 x^{8} - 231 x^{7} + 18000 x^{6} + 29732 x^{5} + 21334 x^{4} + 2464 x^{3} - 9048 x^{2} - 11078 x - 5429 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(24403618437359615692413330078125=5^{15}\cdot 97^{2}\cdot 419^{2}\cdot 695771^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $37.10$
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:  $5, 97, 419, 695771$
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}{10114219932219950522932609961146164096132343741} a^{19} - \frac{3070497870980852131297207783752938979675597610}{10114219932219950522932609961146164096132343741} a^{18} - \frac{4285908751517342846603406035120900622455843889}{10114219932219950522932609961146164096132343741} a^{17} - \frac{4276612934122423354207274919005404721469791904}{10114219932219950522932609961146164096132343741} a^{16} - \frac{3222933128335427143633750438063008056613631020}{10114219932219950522932609961146164096132343741} a^{15} + \frac{1260554864324578957280384045421573939578980684}{10114219932219950522932609961146164096132343741} a^{14} + \frac{400558301357586521885589724065867934198927930}{10114219932219950522932609961146164096132343741} a^{13} + \frac{1475960129728416479185336332650648141366910369}{10114219932219950522932609961146164096132343741} a^{12} + \frac{1381618574188370672734571380819372178403276392}{10114219932219950522932609961146164096132343741} a^{11} - \frac{275559714078991375374040792324594426509828970}{10114219932219950522932609961146164096132343741} a^{10} - \frac{32894116621772503096265309231835876872227523}{10114219932219950522932609961146164096132343741} a^{9} + \frac{1204164132144717481166714645141653567470655586}{10114219932219950522932609961146164096132343741} a^{8} + \frac{2640196351652010605572837754398796430286320832}{10114219932219950522932609961146164096132343741} a^{7} - \frac{2176865471447907828508268110037372481466278193}{10114219932219950522932609961146164096132343741} a^{6} + \frac{1332919314469828023944821920152959113339890327}{10114219932219950522932609961146164096132343741} a^{5} + \frac{3765320374777749287295081275187144500686670646}{10114219932219950522932609961146164096132343741} a^{4} + \frac{1514764846918885935210605348041416538530660391}{10114219932219950522932609961146164096132343741} a^{3} + \frac{1059433334483164482530305922691590086778975382}{10114219932219950522932609961146164096132343741} a^{2} + \frac{1892805184967830210705215965110172893535096533}{10114219932219950522932609961146164096132343741} a + \frac{4095794656803059588290222283888217604731381140}{10114219932219950522932609961146164096132343741}$

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:  $13$
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:  \( 110599474.468 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1039:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 14745600
The 378 conjugacy class representatives for t20n1039 are not computed
Character table for t20n1039 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.10.911025153125.1

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 $20$ $20$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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$5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.12.9.2$x^{12} - 10 x^{8} + 25 x^{4} - 500$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
$97$97.4.2.2$x^{4} - 97 x^{2} + 47045$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
97.8.0.1$x^{8} - x + 84$$1$$8$$0$$C_8$$[\ ]^{8}$
97.8.0.1$x^{8} - x + 84$$1$$8$$0$$C_8$$[\ ]^{8}$
419Data not computed
695771Data not computed