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Properties

Label 20.12.1069756574...3125.2
Degree $20$
Signature $[12, 4]$
Discriminant $5^{15}\cdot 19^{5}\cdot 1699^{5}$
Root discriminant $44.82$
Ramified primes $5, 19, 1699$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T771

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![421, -4576, -3086, 15750, 20529, -11643, -40120, 10118, 21357, -30401, -3883, 13236, -3911, -199, 740, -755, 109, 114, -26, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 - 26*x^18 + 114*x^17 + 109*x^16 - 755*x^15 + 740*x^14 - 199*x^13 - 3911*x^12 + 13236*x^11 - 3883*x^10 - 30401*x^9 + 21357*x^8 + 10118*x^7 - 40120*x^6 - 11643*x^5 + 20529*x^4 + 15750*x^3 - 3086*x^2 - 4576*x + 421)
 
gp: K = bnfinit(x^20 - 4*x^19 - 26*x^18 + 114*x^17 + 109*x^16 - 755*x^15 + 740*x^14 - 199*x^13 - 3911*x^12 + 13236*x^11 - 3883*x^10 - 30401*x^9 + 21357*x^8 + 10118*x^7 - 40120*x^6 - 11643*x^5 + 20529*x^4 + 15750*x^3 - 3086*x^2 - 4576*x + 421, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} - 26 x^{18} + 114 x^{17} + 109 x^{16} - 755 x^{15} + 740 x^{14} - 199 x^{13} - 3911 x^{12} + 13236 x^{11} - 3883 x^{10} - 30401 x^{9} + 21357 x^{8} + 10118 x^{7} - 40120 x^{6} - 11643 x^{5} + 20529 x^{4} + 15750 x^{3} - 3086 x^{2} - 4576 x + 421 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1069756574259831337445098876953125=5^{15}\cdot 19^{5}\cdot 1699^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $44.82$
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:  $5, 19, 1699$
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}{118546291588819958670701122414689837089180701215379} a^{19} - \frac{12786236030261764044531038861530851576489318810421}{118546291588819958670701122414689837089180701215379} a^{18} - \frac{32012334004918084397573427795899964116446507763850}{118546291588819958670701122414689837089180701215379} a^{17} - \frac{7631107140163606132054446825933267872204798214267}{118546291588819958670701122414689837089180701215379} a^{16} + \frac{1382400475680444771910541313023498903078903827951}{118546291588819958670701122414689837089180701215379} a^{15} - \frac{3844847660293427555352419465412545500736919183791}{10776935598983632606427374764971803371743700110489} a^{14} - \frac{53598741533163946689153327437432518103375953124451}{118546291588819958670701122414689837089180701215379} a^{13} + \frac{27926629805669275261534169884118859872340358821103}{118546291588819958670701122414689837089180701215379} a^{12} + \frac{33135742579743820662216323765607920422980550621324}{118546291588819958670701122414689837089180701215379} a^{11} - \frac{49536138435462834123528314312971794192125139376775}{118546291588819958670701122414689837089180701215379} a^{10} - \frac{23040635941272467581101171287257707363649681785754}{118546291588819958670701122414689837089180701215379} a^{9} - \frac{17401125093327683337504372805764690548720752546010}{118546291588819958670701122414689837089180701215379} a^{8} + \frac{56980493458861674426755819207637170765541159488944}{118546291588819958670701122414689837089180701215379} a^{7} + \frac{31812735868891532750990323649828763450035939018112}{118546291588819958670701122414689837089180701215379} a^{6} + \frac{3303967874018132380405706371297939404634003125714}{118546291588819958670701122414689837089180701215379} a^{5} - \frac{19847352289891562509751016712119194080273147224335}{118546291588819958670701122414689837089180701215379} a^{4} - \frac{44881092377195703250901024955758622515354482991903}{118546291588819958670701122414689837089180701215379} a^{3} - \frac{22854350158806193641924715524181677279791469283172}{118546291588819958670701122414689837089180701215379} a^{2} + \frac{8642650747302450776388414659567896611636447298544}{118546291588819958670701122414689837089180701215379} a - \frac{49166827013114763710167330282921217012304331075118}{118546291588819958670701122414689837089180701215379}$

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:  $15$
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:  \( 1922663374.17 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T771:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 102400
The 130 conjugacy class representatives for t20n771 are not computed
Character table for t20n771 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.10.3256446753125.1

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 $20$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ R $20$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{3}$ $20$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
5Data not computed
$19$$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
19.4.0.1$x^{4} - 2 x + 10$$1$$4$$0$$C_4$$[\ ]^{4}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.0.1$x^{4} - 2 x + 10$$1$$4$$0$$C_4$$[\ ]^{4}$
19.4.3.2$x^{4} - 19$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
1699Data not computed