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Properties

Label 20.20.1435256732...1393.1
Degree $20$
Signature $[20, 0]$
Discriminant $13^{10}\cdot 439^{3}\cdot 23087^{3}$
Root discriminant $40.54$
Ramified primes $13, 439, 23087$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T540

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-3, 16, 98, -368, -992, 2055, 3887, -5401, -7306, 7984, 7046, -6993, -3390, 3554, 675, -967, -3, 125, -13, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 - 13*x^18 + 125*x^17 - 3*x^16 - 967*x^15 + 675*x^14 + 3554*x^13 - 3390*x^12 - 6993*x^11 + 7046*x^10 + 7984*x^9 - 7306*x^8 - 5401*x^7 + 3887*x^6 + 2055*x^5 - 992*x^4 - 368*x^3 + 98*x^2 + 16*x - 3)
 
gp: K = bnfinit(x^20 - 6*x^19 - 13*x^18 + 125*x^17 - 3*x^16 - 967*x^15 + 675*x^14 + 3554*x^13 - 3390*x^12 - 6993*x^11 + 7046*x^10 + 7984*x^9 - 7306*x^8 - 5401*x^7 + 3887*x^6 + 2055*x^5 - 992*x^4 - 368*x^3 + 98*x^2 + 16*x - 3, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} - 13 x^{18} + 125 x^{17} - 3 x^{16} - 967 x^{15} + 675 x^{14} + 3554 x^{13} - 3390 x^{12} - 6993 x^{11} + 7046 x^{10} + 7984 x^{9} - 7306 x^{8} - 5401 x^{7} + 3887 x^{6} + 2055 x^{5} - 992 x^{4} - 368 x^{3} + 98 x^{2} + 16 x - 3 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[20, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(143525673214602337250007118901393=13^{10}\cdot 439^{3}\cdot 23087^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $40.54$
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:  $13, 439, 23087$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{69557749065109} a^{19} + \frac{33729633663905}{69557749065109} a^{18} - \frac{11959679194694}{69557749065109} a^{17} + \frac{30096424454413}{69557749065109} a^{16} + \frac{27404938387379}{69557749065109} a^{15} - \frac{10061796431369}{69557749065109} a^{14} + \frac{14406918331374}{69557749065109} a^{13} - \frac{11692927203524}{69557749065109} a^{12} - \frac{32421358715153}{69557749065109} a^{11} + \frac{9785594237913}{69557749065109} a^{10} - \frac{266690318261}{2243798356939} a^{9} - \frac{8954367860571}{69557749065109} a^{8} + \frac{19650498168811}{69557749065109} a^{7} - \frac{17081031187554}{69557749065109} a^{6} + \frac{5941168385791}{69557749065109} a^{5} + \frac{15383699028781}{69557749065109} a^{4} + \frac{11935137988358}{69557749065109} a^{3} + \frac{7125849691739}{69557749065109} a^{2} - \frac{26444489991397}{69557749065109} a + \frac{6797099991506}{69557749065109}$

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:  $19$
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:  \( 5190184029.33 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T540:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 28800
The 35 conjugacy class representatives for t20n540
Character table for t20n540 is not computed

Intermediate fields

\(\Q(\sqrt{13}) \)

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 10 sibling: data not computed
Degree 12 sibling: data not computed
Degree 20 siblings: data not computed
Degree 24 siblings: data not computed
Degree 25 sibling: data not computed
Degree 30 sibling: data not computed
Degree 36 sibling: 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.6.0.1}{6} }{,}\,{\href{/LocalNumberField/3.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ ${\href{/LocalNumberField/5.12.0.1}{12} }{,}\,{\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$ ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.6.0.1}{6} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }$ R ${\href{/LocalNumberField/17.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }$ ${\href{/LocalNumberField/41.12.0.1}{12} }{,}\,{\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }$

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
$13$13.10.5.1$x^{10} - 676 x^{6} + 114244 x^{2} - 13366548$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
13.10.5.1$x^{10} - 676 x^{6} + 114244 x^{2} - 13366548$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
439Data not computed
23087Data not computed