↑

Properties

Label 20.20.7283576068...0625.1
Degree $20$
Signature $[20, 0]$
Discriminant $3^{6}\cdot 5^{16}\cdot 23^{6}\cdot 89^{7}$
Root discriminant $62.10$
Ramified primes $3, 5, 23, 89$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 20T466

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-3231351, -13744080, -11465334, 19787211, 27746442, -9053970, -22303995, 631491, 9278956, 761812, -2282540, -282690, 350741, 45691, -34113, -3905, 2045, 171, -69, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 - 69*x^18 + 171*x^17 + 2045*x^16 - 3905*x^15 - 34113*x^14 + 45691*x^13 + 350741*x^12 - 282690*x^11 - 2282540*x^10 + 761812*x^9 + 9278956*x^8 + 631491*x^7 - 22303995*x^6 - 9053970*x^5 + 27746442*x^4 + 19787211*x^3 - 11465334*x^2 - 13744080*x - 3231351)
 
gp: K = bnfinit(x^20 - 3*x^19 - 69*x^18 + 171*x^17 + 2045*x^16 - 3905*x^15 - 34113*x^14 + 45691*x^13 + 350741*x^12 - 282690*x^11 - 2282540*x^10 + 761812*x^9 + 9278956*x^8 + 631491*x^7 - 22303995*x^6 - 9053970*x^5 + 27746442*x^4 + 19787211*x^3 - 11465334*x^2 - 13744080*x - 3231351, 1)
 

Normalized defining polynomial

\( x^{20} - 3 x^{19} - 69 x^{18} + 171 x^{17} + 2045 x^{16} - 3905 x^{15} - 34113 x^{14} + 45691 x^{13} + 350741 x^{12} - 282690 x^{11} - 2282540 x^{10} + 761812 x^{9} + 9278956 x^{8} + 631491 x^{7} - 22303995 x^{6} - 9053970 x^{5} + 27746442 x^{4} + 19787211 x^{3} - 11465334 x^{2} - 13744080 x - 3231351 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[20, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(728357606894840197718025054931640625=3^{6}\cdot 5^{16}\cdot 23^{6}\cdot 89^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $62.10$
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:  $3, 5, 23, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{15} a^{16} - \frac{1}{5} a^{14} - \frac{2}{5} a^{13} - \frac{7}{15} a^{12} + \frac{7}{15} a^{11} - \frac{1}{5} a^{10} + \frac{1}{15} a^{9} + \frac{2}{15} a^{8} - \frac{2}{5} a^{7} + \frac{4}{15} a^{6} + \frac{1}{15} a^{5} + \frac{4}{15} a^{4} + \frac{1}{5} a^{3} + \frac{2}{5} a^{2} + \frac{1}{5} a - \frac{2}{5}$, $\frac{1}{15} a^{17} - \frac{1}{5} a^{15} - \frac{2}{5} a^{14} - \frac{7}{15} a^{13} + \frac{7}{15} a^{12} - \frac{1}{5} a^{11} + \frac{1}{15} a^{10} + \frac{2}{15} a^{9} - \frac{2}{5} a^{8} + \frac{4}{15} a^{7} + \frac{1}{15} a^{6} + \frac{4}{15} a^{5} + \frac{1}{5} a^{4} + \frac{2}{5} a^{3} + \frac{1}{5} a^{2} - \frac{2}{5} a$, $\frac{1}{1455} a^{18} + \frac{4}{1455} a^{17} + \frac{1}{1455} a^{16} - \frac{111}{485} a^{15} + \frac{47}{1455} a^{14} + \frac{32}{97} a^{13} + \frac{204}{485} a^{12} + \frac{392}{1455} a^{11} - \frac{27}{485} a^{10} + \frac{212}{485} a^{9} - \frac{49}{485} a^{8} - \frac{727}{1455} a^{7} - \frac{122}{485} a^{6} + \frac{293}{1455} a^{5} - \frac{41}{1455} a^{4} - \frac{32}{485} a^{3} + \frac{46}{97} a^{2} - \frac{94}{485} a + \frac{197}{485}$, $\frac{1}{6340818999281159183002215078221819481875025} a^{19} - \frac{607180861977136337424916735099381572087}{2113606333093719727667405026073939827291675} a^{18} + \frac{14899458367051845470753453247085661515173}{2113606333093719727667405026073939827291675} a^{17} + \frac{183135199744448521123916361714136201171184}{6340818999281159183002215078221819481875025} a^{16} - \frac{210956392342973817832923052912702838584117}{6340818999281159183002215078221819481875025} a^{15} - \frac{596474016280852354326137760158544441747314}{6340818999281159183002215078221819481875025} a^{14} - \frac{1016606282040203649931166796987213990210082}{2113606333093719727667405026073939827291675} a^{13} - \frac{2880408390485746550624950993986782809959481}{6340818999281159183002215078221819481875025} a^{12} - \frac{1908479669359665652664143398390081084102251}{6340818999281159183002215078221819481875025} a^{11} + \frac{418918899754686731207401818573908108618106}{2113606333093719727667405026073939827291675} a^{10} - \frac{574771480242085239740286975145365988469783}{2113606333093719727667405026073939827291675} a^{9} - \frac{2402724094696724777562251323172817180876481}{6340818999281159183002215078221819481875025} a^{8} + \frac{831846149142113251905596186354840605960384}{6340818999281159183002215078221819481875025} a^{7} - \frac{555818512696697830931645305485627119116531}{6340818999281159183002215078221819481875025} a^{6} - \frac{3094376504951754871068127074374925378841267}{6340818999281159183002215078221819481875025} a^{5} + \frac{3084133154311057174849725086211173456360711}{6340818999281159183002215078221819481875025} a^{4} - \frac{463853405498297205782056469903994289431612}{2113606333093719727667405026073939827291675} a^{3} + \frac{249158777432785627540926933869332682819373}{2113606333093719727667405026073939827291675} a^{2} - \frac{47535345215739787643111886072206819881902}{2113606333093719727667405026073939827291675} a + \frac{654353587937495046390882680199906492128761}{2113606333093719727667405026073939827291675}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $19$
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:  \( 208796920302 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T466:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 15360
The 90 conjugacy class representatives for t20n466 are not computed
Character table for t20n466 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.767625.1, 10.10.2946240703125.1

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 R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }^{2}$ R ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }^{2}$

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
$3$3.8.6.1$x^{8} + 9 x^{4} + 36$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
3.12.0.1$x^{12} - x^{4} - x^{3} - x^{2} + x - 1$$1$$12$$0$$C_{12}$$[\ ]^{12}$
$5$5.6.5.1$x^{6} - 5$$6$$1$$5$$D_{6}$$[\ ]_{6}^{2}$
5.6.5.1$x^{6} - 5$$6$$1$$5$$D_{6}$$[\ ]_{6}^{2}$
5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$23$23.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
23.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
23.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
23.8.6.1$x^{8} + 299 x^{4} + 25921$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
$89$89.4.2.1$x^{4} + 979 x^{2} + 285156$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
89.4.2.1$x^{4} + 979 x^{2} + 285156$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
89.6.3.2$x^{6} - 7921 x^{2} + 4934783$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
89.6.0.1$x^{6} - x + 6$$1$$6$$0$$C_6$$[\ ]^{6}$