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Properties

Label 20.12.8294330590...3125.1
Degree $20$
Signature $[12, 4]$
Discriminant $5^{10}\cdot 13^{4}\cdot 29^{5}\cdot 347^{4}$
Root discriminant $27.92$
Ramified primes $5, 13, 29, 347$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T174

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, -14, 7, 52, -62, -77, 198, 179, -207, -374, 106, 327, -99, -91, 62, 13, -14, -7, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 7*x^18 - 14*x^17 + 13*x^16 + 62*x^15 - 91*x^14 - 99*x^13 + 327*x^12 + 106*x^11 - 374*x^10 - 207*x^9 + 179*x^8 + 198*x^7 - 77*x^6 - 62*x^5 + 52*x^4 + 7*x^3 - 14*x^2 + 1)
 
gp: K = bnfinit(x^20 - 7*x^18 - 14*x^17 + 13*x^16 + 62*x^15 - 91*x^14 - 99*x^13 + 327*x^12 + 106*x^11 - 374*x^10 - 207*x^9 + 179*x^8 + 198*x^7 - 77*x^6 - 62*x^5 + 52*x^4 + 7*x^3 - 14*x^2 + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 7 x^{18} - 14 x^{17} + 13 x^{16} + 62 x^{15} - 91 x^{14} - 99 x^{13} + 327 x^{12} + 106 x^{11} - 374 x^{10} - 207 x^{9} + 179 x^{8} + 198 x^{7} - 77 x^{6} - 62 x^{5} + 52 x^{4} + 7 x^{3} - 14 x^{2} + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(82943305909096044672939453125=5^{10}\cdot 13^{4}\cdot 29^{5}\cdot 347^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.92$
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:  $5, 13, 29, 347$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{723931} a^{18} + \frac{194234}{723931} a^{17} + \frac{336558}{723931} a^{16} - \frac{852}{3107} a^{15} + \frac{160174}{723931} a^{14} - \frac{245497}{723931} a^{13} + \frac{328564}{723931} a^{12} + \frac{309632}{723931} a^{11} + \frac{10557}{723931} a^{10} + \frac{71307}{723931} a^{9} - \frac{143547}{723931} a^{8} - \frac{38089}{723931} a^{7} - \frac{66036}{723931} a^{6} + \frac{117456}{723931} a^{5} + \frac{121034}{723931} a^{4} + \frac{112841}{723931} a^{3} - \frac{63190}{723931} a^{2} + \frac{117888}{723931} a + \frac{228577}{723931}$, $\frac{1}{1475558876129} a^{19} - \frac{874207}{1475558876129} a^{18} + \frac{576618176151}{1475558876129} a^{17} - \frac{519306221814}{1475558876129} a^{16} + \frac{516158157316}{1475558876129} a^{15} + \frac{712512286575}{1475558876129} a^{14} - \frac{17237047450}{113504528933} a^{13} + \frac{501018743632}{1475558876129} a^{12} - \frac{181763224531}{1475558876129} a^{11} - \frac{136296552582}{1475558876129} a^{10} - \frac{480093908419}{1475558876129} a^{9} - \frac{314903305212}{1475558876129} a^{8} + \frac{425751786389}{1475558876129} a^{7} - \frac{8137714201}{113504528933} a^{6} - \frac{133975536727}{1475558876129} a^{5} + \frac{21404497967}{113504528933} a^{4} - \frac{283313837905}{1475558876129} a^{3} - \frac{34176864985}{113504528933} a^{2} + \frac{671500907984}{1475558876129} a + \frac{162820025258}{1475558876129}$

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:  \( 16712140.3216 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T174:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 960
The 35 conjugacy class representatives for t20n174
Character table for t20n174 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.725.1, 5.3.4511.1, 10.6.63591003125.1

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 24 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 $20$ $20$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ $20$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
5Data not computed
$13$13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{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}$
$29$29.10.5.2$x^{10} - 707281 x^{2} + 225622639$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
29.10.0.1$x^{10} + x^{2} - 2 x + 2$$1$$10$$0$$C_{10}$$[\ ]^{10}$
347Data not computed