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Properties

Label 20.12.2776853807...0000.1
Degree $20$
Signature $[12, 4]$
Discriminant $2^{20}\cdot 5^{10}\cdot 11^{2}\cdot 41^{4}\cdot 28162171^{2}$
Root discriminant $66.40$
Ramified primes $2, 5, 11, 41, 28162171$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1030

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2825761, 0, -3652813, 0, -36982, 0, 1383668, 0, -235376, 0, -146505, 0, 27898, 0, 3608, 0, -378, 0, -13, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 13*x^18 - 378*x^16 + 3608*x^14 + 27898*x^12 - 146505*x^10 - 235376*x^8 + 1383668*x^6 - 36982*x^4 - 3652813*x^2 + 2825761)
 
gp: K = bnfinit(x^20 - 13*x^18 - 378*x^16 + 3608*x^14 + 27898*x^12 - 146505*x^10 - 235376*x^8 + 1383668*x^6 - 36982*x^4 - 3652813*x^2 + 2825761, 1)
 

Normalized defining polynomial

\( x^{20} - 13 x^{18} - 378 x^{16} + 3608 x^{14} + 27898 x^{12} - 146505 x^{10} - 235376 x^{8} + 1383668 x^{6} - 36982 x^{4} - 3652813 x^{2} + 2825761 \)

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:  \(2776853807987152620436695040000000000=2^{20}\cdot 5^{10}\cdot 11^{2}\cdot 41^{4}\cdot 28162171^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $66.40$
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:  $2, 5, 11, 41, 28162171$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{41} a^{14} - \frac{13}{41} a^{12} - \frac{9}{41} a^{10} + \frac{18}{41} a^{6} - \frac{12}{41} a^{4} + \frac{5}{41} a^{2}$, $\frac{1}{41} a^{15} - \frac{13}{41} a^{13} - \frac{9}{41} a^{11} + \frac{18}{41} a^{7} - \frac{12}{41} a^{5} + \frac{5}{41} a^{3}$, $\frac{1}{1681} a^{16} - \frac{13}{1681} a^{14} - \frac{378}{1681} a^{12} + \frac{6}{41} a^{10} - \frac{679}{1681} a^{8} - \frac{258}{1681} a^{6} - \frac{36}{1681} a^{4} + \frac{5}{41} a^{2}$, $\frac{1}{1681} a^{17} - \frac{13}{1681} a^{15} - \frac{378}{1681} a^{13} + \frac{6}{41} a^{11} - \frac{679}{1681} a^{9} - \frac{258}{1681} a^{7} - \frac{36}{1681} a^{5} + \frac{5}{41} a^{3}$, $\frac{1}{22907347082574486283304534119} a^{18} + \frac{3802823569806734634421334}{22907347082574486283304534119} a^{16} - \frac{9718741493974677140392373}{22907347082574486283304534119} a^{14} - \frac{175011334067623806760461687}{558715782501816738617183759} a^{12} - \frac{4575673842423781144442430017}{22907347082574486283304534119} a^{10} - \frac{1833011378892091970591003719}{22907347082574486283304534119} a^{8} + \frac{779884409797765881696606423}{22907347082574486283304534119} a^{6} - \frac{30411452787983456063305822}{558715782501816738617183759} a^{4} + \frac{6676918146309221194019428}{13627214207361383868711799} a^{2} + \frac{80279515782690539792916}{332371078228326435822239}$, $\frac{1}{22907347082574486283304534119} a^{19} + \frac{3802823569806734634421334}{22907347082574486283304534119} a^{17} - \frac{9718741493974677140392373}{22907347082574486283304534119} a^{15} - \frac{175011334067623806760461687}{558715782501816738617183759} a^{13} - \frac{4575673842423781144442430017}{22907347082574486283304534119} a^{11} - \frac{1833011378892091970591003719}{22907347082574486283304534119} a^{9} + \frac{779884409797765881696606423}{22907347082574486283304534119} a^{7} - \frac{30411452787983456063305822}{558715782501816738617183759} a^{5} + \frac{6676918146309221194019428}{13627214207361383868711799} a^{3} + \frac{80279515782690539792916}{332371078228326435822239} a$

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:  \( 73289635003.9 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1030:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7372800
The 189 conjugacy class representatives for t20n1030 are not computed
Character table for t20n1030 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.10.968074628125.1

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 32 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 R ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.8.0.1}{8} }$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{3}$ R ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$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$11.3.0.1$x^{3} - x + 3$$1$$3$$0$$C_3$$[\ ]^{3}$
11.3.0.1$x^{3} - x + 3$$1$$3$$0$$C_3$$[\ ]^{3}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.5.0.1$x^{5} + x^{2} - x + 5$$1$$5$$0$$C_5$$[\ ]^{5}$
11.5.0.1$x^{5} + x^{2} - x + 5$$1$$5$$0$$C_5$$[\ ]^{5}$
$41$$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.3.0.1$x^{3} - x + 13$$1$$3$$0$$C_3$$[\ ]^{3}$
41.3.0.1$x^{3} - x + 13$$1$$3$$0$$C_3$$[\ ]^{3}$
41.8.4.1$x^{8} + 57154 x^{4} - 68921 x^{2} + 816644929$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
28162171Data not computed