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Properties

Label 20.12.2383231946...0000.1
Degree $20$
Signature $[12, 4]$
Discriminant $2^{20}\cdot 5^{10}\cdot 11^{11}\cdot 13^{8}$
Root discriminant $46.65$
Ramified primes $2, 5, 11, 13$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T658

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1331, 0, -18271, 0, 36971, 0, -16172, 0, -5603, 0, 103, 0, 963, 0, 89, 0, -58, 0, -3, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^18 - 58*x^16 + 89*x^14 + 963*x^12 + 103*x^10 - 5603*x^8 - 16172*x^6 + 36971*x^4 - 18271*x^2 + 1331)
 
gp: K = bnfinit(x^20 - 3*x^18 - 58*x^16 + 89*x^14 + 963*x^12 + 103*x^10 - 5603*x^8 - 16172*x^6 + 36971*x^4 - 18271*x^2 + 1331, 1)
 

Normalized defining polynomial

\( x^{20} - 3 x^{18} - 58 x^{16} + 89 x^{14} + 963 x^{12} + 103 x^{10} - 5603 x^{8} - 16172 x^{6} + 36971 x^{4} - 18271 x^{2} + 1331 \)

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:  \(2383231946518789535037440000000000=2^{20}\cdot 5^{10}\cdot 11^{11}\cdot 13^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $46.65$
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:  $2, 5, 11, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{11} a^{12} - \frac{2}{11} a^{10} + \frac{3}{11} a^{8} - \frac{1}{11} a^{6} - \frac{4}{11} a^{4} + \frac{3}{11} a^{2}$, $\frac{1}{11} a^{13} - \frac{2}{11} a^{11} + \frac{3}{11} a^{9} - \frac{1}{11} a^{7} - \frac{4}{11} a^{5} + \frac{3}{11} a^{3}$, $\frac{1}{11} a^{14} - \frac{1}{11} a^{10} + \frac{5}{11} a^{8} + \frac{5}{11} a^{6} - \frac{5}{11} a^{4} - \frac{5}{11} a^{2}$, $\frac{1}{11} a^{15} - \frac{1}{11} a^{11} + \frac{5}{11} a^{9} + \frac{5}{11} a^{7} - \frac{5}{11} a^{5} - \frac{5}{11} a^{3}$, $\frac{1}{11} a^{16} + \frac{3}{11} a^{10} - \frac{3}{11} a^{8} + \frac{5}{11} a^{6} + \frac{2}{11} a^{4} + \frac{3}{11} a^{2}$, $\frac{1}{11} a^{17} + \frac{3}{11} a^{11} - \frac{3}{11} a^{9} + \frac{5}{11} a^{7} + \frac{2}{11} a^{5} + \frac{3}{11} a^{3}$, $\frac{1}{298276875417865421} a^{18} - \frac{4750410124971558}{298276875417865421} a^{16} + \frac{10791308603789799}{298276875417865421} a^{14} - \frac{12190161608957056}{298276875417865421} a^{12} + \frac{2588900648184761}{15698782916729759} a^{10} - \frac{92863143997624040}{298276875417865421} a^{8} + \frac{15788369938657185}{42610982202552203} a^{6} + \frac{141809047430780420}{298276875417865421} a^{4} - \frac{1514506426702274}{27116079583442311} a^{2} - \frac{103019048411683}{2465098143949301}$, $\frac{1}{298276875417865421} a^{19} - \frac{4750410124971558}{298276875417865421} a^{17} + \frac{10791308603789799}{298276875417865421} a^{15} - \frac{12190161608957056}{298276875417865421} a^{13} + \frac{2588900648184761}{15698782916729759} a^{11} - \frac{92863143997624040}{298276875417865421} a^{9} + \frac{15788369938657185}{42610982202552203} a^{7} + \frac{141809047430780420}{298276875417865421} a^{5} - \frac{1514506426702274}{27116079583442311} a^{3} - \frac{103019048411683}{2465098143949301} a$

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:  \( 2943028417.19 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T658:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 57600
The 76 conjugacy class representatives for t20n658 are not computed
Character table for t20n658 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.4400.1, 10.6.1306755003125.1

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 R $20$ R ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ R R ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}$ $20$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
2Data not computed
$5$5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$11$11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.6.3.2$x^{6} - 121 x^{2} + 3993$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
11.8.6.2$x^{8} - 781 x^{4} + 290521$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
$13$13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.12.8.1$x^{12} - 39 x^{9} - 338 x^{6} + 10985 x^{3} + 228488$$3$$4$$8$$C_{12}$$[\ ]_{3}^{4}$