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Properties

Label 20.4.18805933270...1721.1
Degree $20$
Signature $[4, 8]$
Discriminant $13^{10}\cdot 97^{2}\cdot 347^{4}$
Root discriminant $18.35$
Ramified primes $13, 97, 347$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T368

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 2, 12, -113, 348, -518, -12, 1559, -3176, 3440, -2292, 1065, -556, 401, -194, 9, 37, -17, 5, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 + 5*x^18 - 17*x^17 + 37*x^16 + 9*x^15 - 194*x^14 + 401*x^13 - 556*x^12 + 1065*x^11 - 2292*x^10 + 3440*x^9 - 3176*x^8 + 1559*x^7 - 12*x^6 - 518*x^5 + 348*x^4 - 113*x^3 + 12*x^2 + 2*x + 1)
 
gp: K = bnfinit(x^20 - 3*x^19 + 5*x^18 - 17*x^17 + 37*x^16 + 9*x^15 - 194*x^14 + 401*x^13 - 556*x^12 + 1065*x^11 - 2292*x^10 + 3440*x^9 - 3176*x^8 + 1559*x^7 - 12*x^6 - 518*x^5 + 348*x^4 - 113*x^3 + 12*x^2 + 2*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 3 x^{19} + 5 x^{18} - 17 x^{17} + 37 x^{16} + 9 x^{15} - 194 x^{14} + 401 x^{13} - 556 x^{12} + 1065 x^{11} - 2292 x^{10} + 3440 x^{9} - 3176 x^{8} + 1559 x^{7} - 12 x^{6} - 518 x^{5} + 348 x^{4} - 113 x^{3} + 12 x^{2} + 2 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(18805933270743231481641721=13^{10}\cdot 97^{2}\cdot 347^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.35$
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:  $13, 97, 347$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{266719377789090255268411} a^{19} - \frac{74681362344759665676809}{266719377789090255268411} a^{18} - \frac{128736330355431039894194}{266719377789090255268411} a^{17} + \frac{255139929180694329338}{266719377789090255268411} a^{16} - \frac{11993325611220959710912}{266719377789090255268411} a^{15} + \frac{70284675248438030002516}{266719377789090255268411} a^{14} + \frac{21263461784460447563910}{266719377789090255268411} a^{13} - \frac{37091862427819854076597}{266719377789090255268411} a^{12} - \frac{52813121870302787368388}{266719377789090255268411} a^{11} + \frac{69591341090601349475960}{266719377789090255268411} a^{10} - \frac{57380836342982516115059}{266719377789090255268411} a^{9} + \frac{53029745355018149386224}{266719377789090255268411} a^{8} - \frac{93283467426964407711318}{266719377789090255268411} a^{7} + \frac{105126646684966210316902}{266719377789090255268411} a^{6} - \frac{56413633467262946688964}{266719377789090255268411} a^{5} + \frac{73075868396358783563149}{266719377789090255268411} a^{4} - \frac{103767686795552659795602}{266719377789090255268411} a^{3} - \frac{95214038808847542932653}{266719377789090255268411} a^{2} + \frac{98219748351778158605024}{266719377789090255268411} a - \frac{40037999589932046906612}{266719377789090255268411}$

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:  $11$
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:  \( 38639.3248273 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T368:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7680
The 72 conjugacy class representatives for t20n368 are not computed
Character table for t20n368 is not computed

Intermediate fields

\(\Q(\sqrt{13}) \), 5.3.4511.1, 10.2.1973864737.1, 10.6.44707018837.1, 10.2.4336580827189.2

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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
$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}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$97$97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.4.2.1$x^{4} + 873 x^{2} + 235225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
347Data not computed