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Properties

Label 20.12.7063888397...9408.2
Degree $20$
Signature $[12, 4]$
Discriminant $2^{30}\cdot 3^{15}\cdot 71^{9}$
Root discriminant $43.90$
Ramified primes $2, 3, 71$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T647

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![13, 128, -264, -3442, 3974, 11338, -36024, 57422, -53929, 16374, 23208, -33756, 19048, -2026, -4056, 2794, -828, 88, 18, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 18*x^18 + 88*x^17 - 828*x^16 + 2794*x^15 - 4056*x^14 - 2026*x^13 + 19048*x^12 - 33756*x^11 + 23208*x^10 + 16374*x^9 - 53929*x^8 + 57422*x^7 - 36024*x^6 + 11338*x^5 + 3974*x^4 - 3442*x^3 - 264*x^2 + 128*x + 13)
 
gp: K = bnfinit(x^20 - 8*x^19 + 18*x^18 + 88*x^17 - 828*x^16 + 2794*x^15 - 4056*x^14 - 2026*x^13 + 19048*x^12 - 33756*x^11 + 23208*x^10 + 16374*x^9 - 53929*x^8 + 57422*x^7 - 36024*x^6 + 11338*x^5 + 3974*x^4 - 3442*x^3 - 264*x^2 + 128*x + 13, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 18 x^{18} + 88 x^{17} - 828 x^{16} + 2794 x^{15} - 4056 x^{14} - 2026 x^{13} + 19048 x^{12} - 33756 x^{11} + 23208 x^{10} + 16374 x^{9} - 53929 x^{8} + 57422 x^{7} - 36024 x^{6} + 11338 x^{5} + 3974 x^{4} - 3442 x^{3} - 264 x^{2} + 128 x + 13 \)

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:  \(706388839731582814105670315409408=2^{30}\cdot 3^{15}\cdot 71^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $43.90$
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:  $2, 3, 71$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{108009659231044376987455129787860915852049} a^{19} - \frac{950399492838609266282128971664074169039}{108009659231044376987455129787860915852049} a^{18} - \frac{3665071743128461851614012840000972272345}{108009659231044376987455129787860915852049} a^{17} + \frac{18815489356751440396249439616719563304516}{108009659231044376987455129787860915852049} a^{16} + \frac{20103639867934209798725281591957482953041}{108009659231044376987455129787860915852049} a^{15} + \frac{3743073229477557845229234299306256446662}{108009659231044376987455129787860915852049} a^{14} - \frac{19024949283040383800266039952578608137571}{108009659231044376987455129787860915852049} a^{13} - \frac{27889923292276762437303700839153332758612}{108009659231044376987455129787860915852049} a^{12} - \frac{37015286502856404990272277590429054773265}{108009659231044376987455129787860915852049} a^{11} + \frac{4476814578868407583137111496648501806792}{108009659231044376987455129787860915852049} a^{10} - \frac{14688203485432670549259434652608634817338}{108009659231044376987455129787860915852049} a^{9} - \frac{6457501858445121675179798162002541495860}{108009659231044376987455129787860915852049} a^{8} - \frac{23395570189960787737252839602195038013232}{108009659231044376987455129787860915852049} a^{7} + \frac{44026446434600066364811759003906558241572}{108009659231044376987455129787860915852049} a^{6} + \frac{4004162098361799009932989998458371142}{108009659231044376987455129787860915852049} a^{5} + \frac{11979383096221468571232323445086625906502}{108009659231044376987455129787860915852049} a^{4} - \frac{48447190933858233309151086620037984802572}{108009659231044376987455129787860915852049} a^{3} + \frac{21948713025758520676630708930464898571901}{108009659231044376987455129787860915852049} a^{2} + \frac{13296893058775401215957653254225223452332}{108009659231044376987455129787860915852049} a - \frac{28786996935364206344705318695069201854132}{108009659231044376987455129787860915852049}$

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:  \( 1794901705.67 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T647:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 51200
The 152 conjugacy class representatives for t20n647 are not computed
Character table for t20n647 is not computed

Intermediate fields

\(\Q(\sqrt{3}) \), 10.10.6323239406592.1

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 R $20$ $20$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{4}$ $20$ $20$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
3Data not computed
$71$71.5.0.1$x^{5} - x + 8$$1$$5$$0$$C_5$$[\ ]^{5}$
71.5.0.1$x^{5} - x + 8$$1$$5$$0$$C_5$$[\ ]^{5}$
71.10.9.5$x^{10} - 18176$$10$$1$$9$$C_{10}$$[\ ]_{10}$