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Properties

Label 20.2.12962257948...5232.1
Degree $20$
Signature $[2, 9]$
Discriminant $-\,2^{16}\cdot 3^{27}\cdot 11^{10}$
Root discriminant $25.45$
Ramified primes $2, 3, 11$
Class number $1$
Class group Trivial
Galois group $S_5$ (as 20T35)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3, -21, 123, -453, 834, -651, 71, 1, 106, 105, 273, -963, 700, -151, 131, -153, 42, 3, 5, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 5*x^18 + 3*x^17 + 42*x^16 - 153*x^15 + 131*x^14 - 151*x^13 + 700*x^12 - 963*x^11 + 273*x^10 + 105*x^9 + 106*x^8 + x^7 + 71*x^6 - 651*x^5 + 834*x^4 - 453*x^3 + 123*x^2 - 21*x + 3)
 
gp: K = bnfinit(x^20 - 5*x^19 + 5*x^18 + 3*x^17 + 42*x^16 - 153*x^15 + 131*x^14 - 151*x^13 + 700*x^12 - 963*x^11 + 273*x^10 + 105*x^9 + 106*x^8 + x^7 + 71*x^6 - 651*x^5 + 834*x^4 - 453*x^3 + 123*x^2 - 21*x + 3, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} + 5 x^{18} + 3 x^{17} + 42 x^{16} - 153 x^{15} + 131 x^{14} - 151 x^{13} + 700 x^{12} - 963 x^{11} + 273 x^{10} + 105 x^{9} + 106 x^{8} + x^{7} + 71 x^{6} - 651 x^{5} + 834 x^{4} - 453 x^{3} + 123 x^{2} - 21 x + 3 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-12962257948142832318230495232=-\,2^{16}\cdot 3^{27}\cdot 11^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $25.45$
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:  $2, 3, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{209} a^{18} + \frac{29}{209} a^{17} + \frac{75}{209} a^{16} + \frac{24}{209} a^{15} + \frac{83}{209} a^{14} - \frac{87}{209} a^{13} - \frac{62}{209} a^{12} + \frac{103}{209} a^{11} - \frac{34}{209} a^{10} + \frac{91}{209} a^{9} + \frac{26}{209} a^{8} - \frac{21}{209} a^{7} + \frac{29}{209} a^{6} - \frac{50}{209} a^{5} + \frac{2}{19} a^{4} - \frac{83}{209} a^{3} + \frac{14}{209} a^{2} - \frac{25}{209} a + \frac{34}{209}$, $\frac{1}{29314599977024924903} a^{19} - \frac{1601853246826048}{2664963634274993173} a^{18} + \frac{372065296051851661}{1542873683001311837} a^{17} + \frac{2433570591978299471}{29314599977024924903} a^{16} - \frac{1303937665926497005}{29314599977024924903} a^{15} + \frac{463701093170769413}{29314599977024924903} a^{14} + \frac{13974176254950732950}{29314599977024924903} a^{13} - \frac{9855543525034642305}{29314599977024924903} a^{12} + \frac{45988916416932343}{480567212738113523} a^{11} - \frac{11491752303441911887}{29314599977024924903} a^{10} + \frac{8867634478775736922}{29314599977024924903} a^{9} - \frac{11400043620493308973}{29314599977024924903} a^{8} - \frac{195989289477971553}{2664963634274993173} a^{7} + \frac{662783256871257316}{2664963634274993173} a^{6} - \frac{4294927420302061215}{29314599977024924903} a^{5} + \frac{9271694761804858729}{29314599977024924903} a^{4} + \frac{479759338328092874}{1542873683001311837} a^{3} - \frac{8596913230565388324}{29314599977024924903} a^{2} - \frac{778825625687428384}{2664963634274993173} a + \frac{7290451386776511518}{29314599977024924903}$

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

Class group and class number

Trivial group, which has order $1$

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $10$
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 1605879.29863 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_5$ (as 20T35):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 120
The 7 conjugacy class representatives for $S_5$
Character table for $S_5$

Intermediate fields

10.4.5975675659008.1

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

Sibling fields

Degree 5 sibling: data not computed
Degree 6 sibling: data not computed
Degree 10 siblings: data not computed
Degree 12 sibling: data not computed
Degree 15 sibling: data not computed
Degree 20 siblings: data not computed
Degree 24 sibling: data not computed
Degree 30 siblings: data not computed
Degree 40 sibling: 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 R ${\href{/LocalNumberField/5.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$3$3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.6.9.4$x^{6} + 6 x^{4} + 6$$6$$1$$9$$D_{6}$$[2]_{2}^{2}$
3.12.18.85$x^{12} + 36 x^{11} + 111 x^{10} + 90 x^{9} + 36 x^{8} + 90 x^{7} + 30 x^{6} + 108 x^{5} - 36 x^{4} + 54 x^{3} - 81 x^{2} + 54 x - 18$$6$$2$$18$$D_6$$[2]_{2}^{2}$
$11$11.4.2.2$x^{4} - 11 x^{2} + 847$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
11.4.2.2$x^{4} - 11 x^{2} + 847$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
11.4.2.2$x^{4} - 11 x^{2} + 847$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
11.4.2.2$x^{4} - 11 x^{2} + 847$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
11.4.2.2$x^{4} - 11 x^{2} + 847$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$