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Properties

Label 20.10.4223146772...0000.2
Degree $20$
Signature $[10, 5]$
Discriminant $-\,2^{30}\cdot 5^{10}\cdot 3319^{5}$
Root discriminant $48.00$
Ramified primes $2, 5, 3319$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 20T756

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-3398656, 0, 5722112, 0, -3109888, 0, 410112, 0, 157632, 0, -50912, 0, -448, 0, 1496, 0, -76, 0, -14, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 14*x^18 - 76*x^16 + 1496*x^14 - 448*x^12 - 50912*x^10 + 157632*x^8 + 410112*x^6 - 3109888*x^4 + 5722112*x^2 - 3398656)
 
gp: K = bnfinit(x^20 - 14*x^18 - 76*x^16 + 1496*x^14 - 448*x^12 - 50912*x^10 + 157632*x^8 + 410112*x^6 - 3109888*x^4 + 5722112*x^2 - 3398656, 1)
 

Normalized defining polynomial

\( x^{20} - 14 x^{18} - 76 x^{16} + 1496 x^{14} - 448 x^{12} - 50912 x^{10} + 157632 x^{8} + 410112 x^{6} - 3109888 x^{4} + 5722112 x^{2} - 3398656 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-4223146772847631783690240000000000=-\,2^{30}\cdot 5^{10}\cdot 3319^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $48.00$
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:  $2, 5, 3319$
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 \(a\))

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{4} a^{4}$, $\frac{1}{4} a^{5}$, $\frac{1}{8} a^{6}$, $\frac{1}{8} a^{7}$, $\frac{1}{16} a^{8}$, $\frac{1}{16} a^{9}$, $\frac{1}{32} a^{10}$, $\frac{1}{32} a^{11}$, $\frac{1}{192} a^{12} + \frac{1}{96} a^{10} + \frac{1}{48} a^{8} + \frac{1}{24} a^{6} + \frac{1}{6} a^{2} + \frac{1}{3}$, $\frac{1}{192} a^{13} + \frac{1}{96} a^{11} + \frac{1}{48} a^{9} + \frac{1}{24} a^{7} + \frac{1}{6} a^{3} + \frac{1}{3} a$, $\frac{1}{384} a^{14} - \frac{1}{24} a^{6} + \frac{1}{12} a^{4} - \frac{1}{3}$, $\frac{1}{384} a^{15} - \frac{1}{24} a^{7} + \frac{1}{12} a^{5} - \frac{1}{3} a$, $\frac{1}{2304} a^{16} + \frac{1}{1152} a^{14} - \frac{1}{144} a^{8} + \frac{1}{24} a^{6} + \frac{1}{36} a^{4} - \frac{2}{9} a^{2} - \frac{1}{9}$, $\frac{1}{2304} a^{17} + \frac{1}{1152} a^{15} - \frac{1}{144} a^{9} + \frac{1}{24} a^{7} + \frac{1}{36} a^{5} - \frac{2}{9} a^{3} - \frac{1}{9} a$, $\frac{1}{8753189322253824} a^{18} + \frac{78578771135}{364716221760576} a^{16} - \frac{921687564865}{2188297330563456} a^{14} + \frac{65906395391}{60786036960096} a^{12} + \frac{2805050934265}{273537166320432} a^{10} + \frac{217393085161}{39076738045776} a^{8} + \frac{5614051554313}{136768583160216} a^{6} - \frac{5512195217789}{68384291580108} a^{4} + \frac{2159302522199}{11397381930018} a^{2} - \frac{4167234040397}{17096072895027}$, $\frac{1}{8753189322253824} a^{19} + \frac{78578771135}{364716221760576} a^{17} - \frac{921687564865}{2188297330563456} a^{15} + \frac{65906395391}{60786036960096} a^{13} + \frac{2805050934265}{273537166320432} a^{11} + \frac{217393085161}{39076738045776} a^{9} + \frac{5614051554313}{136768583160216} a^{7} - \frac{5512195217789}{68384291580108} a^{5} + \frac{2159302522199}{11397381930018} a^{3} - \frac{4167234040397}{17096072895027} a$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}$, which has order $2$ (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:  $14$
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:  \( 1284399186.05 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T756:

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 t20n756 are not computed
Character table for t20n756 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.6.34424253125.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
2Data not computed
5Data not computed
3319Data not computed