↑

Properties

Label 20.6.25454644418...0368.1
Degree $20$
Signature $[6, 7]$
Discriminant $-\,2^{40}\cdot 17^{5}\cdot 439^{5}$
Root discriminant $37.18$
Ramified primes $2, 17, 439$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T797

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 256, -238, 280, 866, -132, 552, -552, -1063, 328, -590, -144, 556, -40, -64, 80, 26, -4, -6, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^18 - 4*x^17 + 26*x^16 + 80*x^15 - 64*x^14 - 40*x^13 + 556*x^12 - 144*x^11 - 590*x^10 + 328*x^9 - 1063*x^8 - 552*x^7 + 552*x^6 - 132*x^5 + 866*x^4 + 280*x^3 - 238*x^2 + 256*x + 1)
 
gp: K = bnfinit(x^20 - 6*x^18 - 4*x^17 + 26*x^16 + 80*x^15 - 64*x^14 - 40*x^13 + 556*x^12 - 144*x^11 - 590*x^10 + 328*x^9 - 1063*x^8 - 552*x^7 + 552*x^6 - 132*x^5 + 866*x^4 + 280*x^3 - 238*x^2 + 256*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{18} - 4 x^{17} + 26 x^{16} + 80 x^{15} - 64 x^{14} - 40 x^{13} + 556 x^{12} - 144 x^{11} - 590 x^{10} + 328 x^{9} - 1063 x^{8} - 552 x^{7} + 552 x^{6} - 132 x^{5} + 866 x^{4} + 280 x^{3} - 238 x^{2} + 256 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-25454644418510559398901598650368=-\,2^{40}\cdot 17^{5}\cdot 439^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $37.18$
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:  $2, 17, 439$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{330307554494235777483772735083515} a^{19} + \frac{100746282598983305811765765068833}{330307554494235777483772735083515} a^{18} - \frac{32425631554953291733612777226187}{330307554494235777483772735083515} a^{17} - \frac{30115434601614686186258349778159}{66061510898847155496754547016703} a^{16} - \frac{160010105361072924596913355327649}{330307554494235777483772735083515} a^{15} - \frac{120203330898028134848824894195867}{330307554494235777483772735083515} a^{14} - \frac{29943256832392132434692191852122}{66061510898847155496754547016703} a^{13} - \frac{26910586472502151701888468995914}{66061510898847155496754547016703} a^{12} - \frac{109285079827861967075240643105574}{330307554494235777483772735083515} a^{11} - \frac{146093406261247104247801706468101}{330307554494235777483772735083515} a^{10} - \frac{13414341840777760599090785744848}{330307554494235777483772735083515} a^{9} - \frac{70713275875187378189145215382351}{330307554494235777483772735083515} a^{8} - \frac{43302754272903315697030363414016}{330307554494235777483772735083515} a^{7} - \frac{18244460248958709870988075546810}{66061510898847155496754547016703} a^{6} - \frac{34720121170525561143774410598068}{330307554494235777483772735083515} a^{5} - \frac{162133093268560824211599034116676}{330307554494235777483772735083515} a^{4} + \frac{79998309077373681730240335833903}{330307554494235777483772735083515} a^{3} - \frac{161501532427561262318771541497631}{330307554494235777483772735083515} a^{2} - \frac{78866780653265287015051694534786}{330307554494235777483772735083515} a - \frac{164886446710775634021474422495032}{330307554494235777483772735083515}$

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:  \( 101166991.924 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T797:

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 t20n797 are not computed
Character table for t20n797 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 5.3.7463.1, 10.6.1825058619392.1

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 R ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.12.0.1}{12} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
2Data not computed
$17$$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.4.3.2$x^{4} - 153$$4$$1$$3$$C_4$$[\ ]_{4}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
439Data not computed