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Properties

Label 20.12.3978052474...5744.1
Degree $20$
Signature $[12, 4]$
Discriminant $2^{16}\cdot 13^{15}\cdot 17^{9}$
Root discriminant $42.66$
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![13, 143, -698, 466, 1828, -3173, 546, 3660, -4329, 620, 2638, -2556, 686, 650, -740, 288, 15, -75, 40, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 40*x^18 - 75*x^17 + 15*x^16 + 288*x^15 - 740*x^14 + 650*x^13 + 686*x^12 - 2556*x^11 + 2638*x^10 + 620*x^9 - 4329*x^8 + 3660*x^7 + 546*x^6 - 3173*x^5 + 1828*x^4 + 466*x^3 - 698*x^2 + 143*x + 13)
 
gp: K = bnfinit(x^20 - 10*x^19 + 40*x^18 - 75*x^17 + 15*x^16 + 288*x^15 - 740*x^14 + 650*x^13 + 686*x^12 - 2556*x^11 + 2638*x^10 + 620*x^9 - 4329*x^8 + 3660*x^7 + 546*x^6 - 3173*x^5 + 1828*x^4 + 466*x^3 - 698*x^2 + 143*x + 13, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 40 x^{18} - 75 x^{17} + 15 x^{16} + 288 x^{15} - 740 x^{14} + 650 x^{13} + 686 x^{12} - 2556 x^{11} + 2638 x^{10} + 620 x^{9} - 4329 x^{8} + 3660 x^{7} + 546 x^{6} - 3173 x^{5} + 1828 x^{4} + 466 x^{3} - 698 x^{2} + 143 x + 13 \)

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:  \(397805247472838231448960252575744=2^{16}\cdot 13^{15}\cdot 17^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $42.66$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{11} a^{16} + \frac{3}{11} a^{15} + \frac{4}{11} a^{14} + \frac{2}{11} a^{13} - \frac{2}{11} a^{12} - \frac{4}{11} a^{11} - \frac{2}{11} a^{10} - \frac{1}{11} a^{9} - \frac{5}{11} a^{8} + \frac{4}{11} a^{7} - \frac{5}{11} a^{6} + \frac{4}{11} a^{5} + \frac{3}{11} a^{4} + \frac{1}{11} a^{3} + \frac{4}{11} a^{2} + \frac{4}{11} a - \frac{1}{11}$, $\frac{1}{11} a^{17} - \frac{5}{11} a^{15} + \frac{1}{11} a^{14} + \frac{3}{11} a^{13} + \frac{2}{11} a^{12} - \frac{1}{11} a^{11} + \frac{5}{11} a^{10} - \frac{2}{11} a^{9} - \frac{3}{11} a^{8} + \frac{5}{11} a^{7} - \frac{3}{11} a^{6} + \frac{2}{11} a^{5} + \frac{3}{11} a^{4} + \frac{1}{11} a^{3} + \frac{3}{11} a^{2} - \frac{2}{11} a + \frac{3}{11}$, $\frac{1}{2679523} a^{18} - \frac{9}{2679523} a^{17} - \frac{44976}{2679523} a^{16} + \frac{360012}{2679523} a^{15} - \frac{21451}{382789} a^{14} + \frac{15603}{382789} a^{13} - \frac{993984}{2679523} a^{12} - \frac{508955}{2679523} a^{11} - \frac{7190}{22517} a^{10} - \frac{101050}{2679523} a^{9} - \frac{1032919}{2679523} a^{8} - \frac{1199852}{2679523} a^{7} + \frac{74901}{243593} a^{6} + \frac{677958}{2679523} a^{5} + \frac{1322110}{2679523} a^{4} - \frac{181418}{382789} a^{3} - \frac{1062014}{2679523} a^{2} + \frac{1246716}{2679523} a + \frac{517284}{2679523}$, $\frac{1}{463557479} a^{19} + \frac{1}{6020227} a^{18} + \frac{593751}{42141589} a^{17} + \frac{542838}{66222497} a^{16} + \frac{62721558}{463557479} a^{15} - \frac{1090159}{3895441} a^{14} - \frac{29601486}{463557479} a^{13} + \frac{164422025}{463557479} a^{12} - \frac{209051015}{463557479} a^{11} - \frac{208146846}{463557479} a^{10} - \frac{22146462}{463557479} a^{9} + \frac{12764372}{42141589} a^{8} + \frac{58895205}{463557479} a^{7} - \frac{172302289}{463557479} a^{6} + \frac{56459789}{463557479} a^{5} + \frac{194278782}{463557479} a^{4} + \frac{50982916}{463557479} a^{3} - \frac{65240002}{463557479} a^{2} - \frac{200410285}{463557479} a - \frac{101182190}{463557479}$

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:  \( 1636849507.33 \) (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 ${\href{/LocalNumberField/3.10.0.1}{10} }{,}\,{\href{/LocalNumberField/3.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ R R ${\href{/LocalNumberField/19.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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$13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.4.3.1$x^{4} - 13$$4$$1$$3$$C_4$$[\ ]_{4}$
13.6.5.3$x^{6} - 208$$6$$1$$5$$C_6$$[\ ]_{6}$
13.6.5.3$x^{6} - 208$$6$$1$$5$$C_6$$[\ ]_{6}$
$17$17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.6.5.1$x^{6} - 17$$6$$1$$5$$D_{6}$$[\ ]_{6}^{2}$
17.6.4.1$x^{6} + 136 x^{3} + 7803$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$