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Properties

Label 20.4.31602750247...5625.1
Degree $20$
Signature $[4, 8]$
Discriminant $5^{10}\cdot 61^{4}\cdot 97^{2}\cdot 397^{4}$
Root discriminant $26.61$
Ramified primes $5, 61, 97, 397$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T368

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -2, -34, -84, -87, 67, 166, -284, -19, 155, -27, -401, 659, -458, 224, 7, -3, -8, 6, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 6*x^18 - 8*x^17 - 3*x^16 + 7*x^15 + 224*x^14 - 458*x^13 + 659*x^12 - 401*x^11 - 27*x^10 + 155*x^9 - 19*x^8 - 284*x^7 + 166*x^6 + 67*x^5 - 87*x^4 - 84*x^3 - 34*x^2 - 2*x + 1)
 
gp: K = bnfinit(x^20 + 6*x^18 - 8*x^17 - 3*x^16 + 7*x^15 + 224*x^14 - 458*x^13 + 659*x^12 - 401*x^11 - 27*x^10 + 155*x^9 - 19*x^8 - 284*x^7 + 166*x^6 + 67*x^5 - 87*x^4 - 84*x^3 - 34*x^2 - 2*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} + 6 x^{18} - 8 x^{17} - 3 x^{16} + 7 x^{15} + 224 x^{14} - 458 x^{13} + 659 x^{12} - 401 x^{11} - 27 x^{10} + 155 x^{9} - 19 x^{8} - 284 x^{7} + 166 x^{6} + 67 x^{5} - 87 x^{4} - 84 x^{3} - 34 x^{2} - 2 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(31602750247376961471572265625=5^{10}\cdot 61^{4}\cdot 97^{2}\cdot 397^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.61$
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:  $5, 61, 97, 397$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{10417} a^{18} - \frac{2204}{10417} a^{17} - \frac{3823}{10417} a^{16} - \frac{796}{10417} a^{15} - \frac{4901}{10417} a^{14} + \frac{222}{947} a^{13} - \frac{4316}{10417} a^{12} + \frac{3309}{10417} a^{11} + \frac{1824}{10417} a^{10} + \frac{4129}{10417} a^{9} + \frac{1762}{10417} a^{8} - \frac{1428}{10417} a^{7} + \frac{3118}{10417} a^{6} - \frac{2901}{10417} a^{5} - \frac{2538}{10417} a^{4} - \frac{3615}{10417} a^{3} + \frac{277}{947} a^{2} + \frac{2114}{10417} a + \frac{2356}{10417}$, $\frac{1}{2239034156233680351934399} a^{19} - \frac{3931888709423820876}{203548559657607304721309} a^{18} - \frac{85413980759499393241523}{203548559657607304721309} a^{17} + \frac{150610596720467802888271}{2239034156233680351934399} a^{16} + \frac{14569608035697771957415}{203548559657607304721309} a^{15} + \frac{779002541217601686627288}{2239034156233680351934399} a^{14} + \frac{322783905400702871263211}{2239034156233680351934399} a^{13} + \frac{900589922446437274312078}{2239034156233680351934399} a^{12} - \frac{918531275417117156478738}{2239034156233680351934399} a^{11} + \frac{562569561574182065931881}{2239034156233680351934399} a^{10} - \frac{153386967263576974232804}{2239034156233680351934399} a^{9} - \frac{744892461787799058322414}{2239034156233680351934399} a^{8} + \frac{860124171687704532485219}{2239034156233680351934399} a^{7} + \frac{297508112841248114246024}{2239034156233680351934399} a^{6} - \frac{262173714556350102927681}{2239034156233680351934399} a^{5} - \frac{56779812278324471822036}{131707891543157667760847} a^{4} + \frac{66036442448564220105511}{2239034156233680351934399} a^{3} + \frac{53935119415737188207924}{131707891543157667760847} a^{2} + \frac{17625228944827850162815}{131707891543157667760847} a + \frac{876687093896520511243526}{2239034156233680351934399}$

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:  $11$
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:  \( 2432558.68704 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T368:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7680
The 72 conjugacy class representatives for t20n368 are not computed
Character table for t20n368 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.24217.1, 10.2.56886919633.1, 10.10.1832697153125.1, 10.2.177771623853125.1

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 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
5Data not computed
$61$$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
61.2.1.1$x^{2} - 61$$2$$1$$1$$C_2$$[\ ]_{2}$
61.2.1.1$x^{2} - 61$$2$$1$$1$$C_2$$[\ ]_{2}$
61.2.1.1$x^{2} - 61$$2$$1$$1$$C_2$$[\ ]_{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.1.1$x^{2} - 61$$2$$1$$1$$C_2$$[\ ]_{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
$97$97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.4.2.1$x^{4} + 873 x^{2} + 235225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
97.6.0.1$x^{6} - x + 10$$1$$6$$0$$C_6$$[\ ]^{6}$
97.6.0.1$x^{6} - x + 10$$1$$6$$0$$C_6$$[\ ]^{6}$
397Data not computed