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Properties

Label 20.12.6364883959...1904.1
Degree $20$
Signature $[12, 4]$
Discriminant $2^{20}\cdot 13^{15}\cdot 17^{9}$
Root discriminant $49.00$
Ramified primes $2, 13, 17$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T803

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-2323, -8430, 14300, 49930, -8279, -51880, 4914, 4652, -2738, 25938, -2990, -20634, 3922, 7344, -1662, -1348, 329, 120, -30, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 - 30*x^18 + 120*x^17 + 329*x^16 - 1348*x^15 - 1662*x^14 + 7344*x^13 + 3922*x^12 - 20634*x^11 - 2990*x^10 + 25938*x^9 - 2738*x^8 + 4652*x^7 + 4914*x^6 - 51880*x^5 - 8279*x^4 + 49930*x^3 + 14300*x^2 - 8430*x - 2323)
 
gp: K = bnfinit(x^20 - 4*x^19 - 30*x^18 + 120*x^17 + 329*x^16 - 1348*x^15 - 1662*x^14 + 7344*x^13 + 3922*x^12 - 20634*x^11 - 2990*x^10 + 25938*x^9 - 2738*x^8 + 4652*x^7 + 4914*x^6 - 51880*x^5 - 8279*x^4 + 49930*x^3 + 14300*x^2 - 8430*x - 2323, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} - 30 x^{18} + 120 x^{17} + 329 x^{16} - 1348 x^{15} - 1662 x^{14} + 7344 x^{13} + 3922 x^{12} - 20634 x^{11} - 2990 x^{10} + 25938 x^{9} - 2738 x^{8} + 4652 x^{7} + 4914 x^{6} - 51880 x^{5} - 8279 x^{4} + 49930 x^{3} + 14300 x^{2} - 8430 x - 2323 \)

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:  \(6364883959565411703183364041211904=2^{20}\cdot 13^{15}\cdot 17^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $49.00$
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:  $2, 13, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{2} a^{16} - \frac{1}{2}$, $\frac{1}{2} a^{17} - \frac{1}{2} a$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{2}$, $\frac{1}{200935060219013698164885799136983754} a^{19} - \frac{36289202398232468931619450442905279}{200935060219013698164885799136983754} a^{18} - \frac{4138683501409075709854883645784071}{18266823656273972560444163557907614} a^{17} - \frac{20968121319499286377008745183845789}{200935060219013698164885799136983754} a^{16} + \frac{21772576036293409563959956100269775}{100467530109506849082442899568491877} a^{15} + \frac{20919040017038843637718551412758645}{200935060219013698164885799136983754} a^{14} + \frac{20787743429730257262094200401776225}{100467530109506849082442899568491877} a^{13} - \frac{22442017589262558330653531315101523}{200935060219013698164885799136983754} a^{12} - \frac{3936543638971755837955075704819435}{100467530109506849082442899568491877} a^{11} - \frac{30622356824213412441591445801530145}{200935060219013698164885799136983754} a^{10} - \frac{44896283620728799079701517868632980}{100467530109506849082442899568491877} a^{9} + \frac{86040613490995749468526024203930269}{200935060219013698164885799136983754} a^{8} - \frac{32182865011564074929096138504718895}{100467530109506849082442899568491877} a^{7} - \frac{57259803853145911150458923406287823}{200935060219013698164885799136983754} a^{6} + \frac{42490704103849711897045956077772690}{100467530109506849082442899568491877} a^{5} - \frac{33800375824329054359959982211938343}{200935060219013698164885799136983754} a^{4} + \frac{35136555357895379424227098530420715}{200935060219013698164885799136983754} a^{3} + \frac{2644996865366964960253251239854841}{100467530109506849082442899568491877} a^{2} - \frac{30290442031392473797497064658759375}{200935060219013698164885799136983754} a - \frac{22148387463334536464472729531457220}{100467530109506849082442899568491877}$

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:  \( 7110302703.23 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T803:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 122880
The 126 conjugacy class representatives for t20n803 are not computed
Character table for t20n803 is not computed

Intermediate fields

\(\Q(\sqrt{13}) \), 5.5.10158928.1, 10.10.1341649635419392.1

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 ${\href{/LocalNumberField/3.10.0.1}{10} }{,}\,{\href{/LocalNumberField/3.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ R R ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$13$13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.3.1$x^{4} - 13$$4$$1$$3$$C_4$$[\ ]_{4}$
13.12.10.2$x^{12} + 39 x^{6} + 676$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$
$17$17.3.2.1$x^{3} - 17$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
17.3.2.1$x^{3} - 17$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.6.5.1$x^{6} - 17$$6$$1$$5$$D_{6}$$[\ ]_{6}^{2}$