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Properties

Label 20.8.30743450172...2400.1
Degree $20$
Signature $[8, 6]$
Discriminant $2^{40}\cdot 5^{2}\cdot 5783^{4}$
Root discriminant $26.57$
Ramified primes $2, 5, 5783$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T794

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-49, -188, 120, 232, -1856, -276, 4388, -832, -3411, 1232, 810, -16, -287, -104, 88, 0, 21, 0, -10, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^18 + 21*x^16 + 88*x^14 - 104*x^13 - 287*x^12 - 16*x^11 + 810*x^10 + 1232*x^9 - 3411*x^8 - 832*x^7 + 4388*x^6 - 276*x^5 - 1856*x^4 + 232*x^3 + 120*x^2 - 188*x - 49)
 
gp: K = bnfinit(x^20 - 10*x^18 + 21*x^16 + 88*x^14 - 104*x^13 - 287*x^12 - 16*x^11 + 810*x^10 + 1232*x^9 - 3411*x^8 - 832*x^7 + 4388*x^6 - 276*x^5 - 1856*x^4 + 232*x^3 + 120*x^2 - 188*x - 49, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{18} + 21 x^{16} + 88 x^{14} - 104 x^{13} - 287 x^{12} - 16 x^{11} + 810 x^{10} + 1232 x^{9} - 3411 x^{8} - 832 x^{7} + 4388 x^{6} - 276 x^{5} - 1856 x^{4} + 232 x^{3} + 120 x^{2} - 188 x - 49 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(30743450172982969617507942400=2^{40}\cdot 5^{2}\cdot 5783^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.57$
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:  $2, 5, 5783$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{11} a^{18} + \frac{5}{11} a^{17} + \frac{5}{11} a^{16} - \frac{3}{11} a^{15} - \frac{3}{11} a^{13} - \frac{4}{11} a^{12} + \frac{5}{11} a^{11} - \frac{2}{11} a^{10} + \frac{1}{11} a^{9} - \frac{1}{11} a^{8} - \frac{4}{11} a^{7} - \frac{1}{11} a^{4} + \frac{5}{11} a^{3} + \frac{5}{11} a^{2} - \frac{2}{11} a + \frac{5}{11}$, $\frac{1}{323025094378374123484164217667987} a^{19} + \frac{7463841693412351382254635753665}{323025094378374123484164217667987} a^{18} + \frac{72372711439458119834659838222303}{323025094378374123484164217667987} a^{17} - \frac{7695803657966424211920264779783}{29365917670761283953105837969817} a^{16} - \frac{132015010943443379916010196561556}{323025094378374123484164217667987} a^{15} + \frac{49436825825184946292480515515311}{323025094378374123484164217667987} a^{14} + \frac{92566100573599751064457167146308}{323025094378374123484164217667987} a^{13} - \frac{101775302106603071951690523835265}{323025094378374123484164217667987} a^{12} - \frac{90470474367133669143156930312676}{323025094378374123484164217667987} a^{11} + \frac{155377121505526639984268755388419}{323025094378374123484164217667987} a^{10} + \frac{156387805178888600658835754452174}{323025094378374123484164217667987} a^{9} + \frac{14966960668583156983036608752422}{323025094378374123484164217667987} a^{8} + \frac{62960543432967248966820726562002}{323025094378374123484164217667987} a^{7} + \frac{4421651740628302640689173693348}{29365917670761283953105837969817} a^{6} - \frac{21136173094025666461068718431339}{323025094378374123484164217667987} a^{5} - \frac{4167086318546696594239544563944}{29365917670761283953105837969817} a^{4} + \frac{41726639994855583524849803598962}{323025094378374123484164217667987} a^{3} - \frac{111658649498687559342288679318589}{323025094378374123484164217667987} a^{2} - \frac{112322197770339642432083016440561}{323025094378374123484164217667987} a + \frac{160206450468889811931559147790051}{323025094378374123484164217667987}$

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:  $13$
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:  \( 5739160.72004 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T794:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 122880
The 138 conjugacy class representatives for t20n794 are not computed
Character table for t20n794 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 5.3.5783.1, 10.6.1095863140352.2

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 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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5.12.0.1$x^{12} - x^{3} - 2 x + 3$$1$$12$$0$$C_{12}$$[\ ]^{12}$
5783Data not computed