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Properties

Label 20.12.5274403944...0000.1
Degree $20$
Signature $[12, 4]$
Discriminant $2^{40}\cdot 5^{12}\cdot 97\cdot 1193^{4}$
Root discriminant $54.46$
Ramified primes $2, 5, 97, 1193$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T925

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1423, 5376, 140, 19460, 52202, -35332, -94656, -8592, 52793, 11480, -16672, 7340, 11528, -3156, -3874, 400, 587, -16, -40, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 40*x^18 - 16*x^17 + 587*x^16 + 400*x^15 - 3874*x^14 - 3156*x^13 + 11528*x^12 + 7340*x^11 - 16672*x^10 + 11480*x^9 + 52793*x^8 - 8592*x^7 - 94656*x^6 - 35332*x^5 + 52202*x^4 + 19460*x^3 + 140*x^2 + 5376*x + 1423)
 
gp: K = bnfinit(x^20 - 40*x^18 - 16*x^17 + 587*x^16 + 400*x^15 - 3874*x^14 - 3156*x^13 + 11528*x^12 + 7340*x^11 - 16672*x^10 + 11480*x^9 + 52793*x^8 - 8592*x^7 - 94656*x^6 - 35332*x^5 + 52202*x^4 + 19460*x^3 + 140*x^2 + 5376*x + 1423, 1)
 

Normalized defining polynomial

\( x^{20} - 40 x^{18} - 16 x^{17} + 587 x^{16} + 400 x^{15} - 3874 x^{14} - 3156 x^{13} + 11528 x^{12} + 7340 x^{11} - 16672 x^{10} + 11480 x^{9} + 52793 x^{8} - 8592 x^{7} - 94656 x^{6} - 35332 x^{5} + 52202 x^{4} + 19460 x^{3} + 140 x^{2} + 5376 x + 1423 \)

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:  \(52744039446596112351232000000000000=2^{40}\cdot 5^{12}\cdot 97\cdot 1193^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $54.46$
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:  $2, 5, 97, 1193$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{16} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{696896855545530727683606569360695941759529244} a^{19} - \frac{21980989739246964581562947686183432011428489}{696896855545530727683606569360695941759529244} a^{18} + \frac{206554054399712202011518051424125191188334305}{696896855545530727683606569360695941759529244} a^{17} - \frac{95467855800035218925022250153506355352916269}{696896855545530727683606569360695941759529244} a^{16} + \frac{2474241913106947453889178919458924609823310}{24889173412340383131557377477167712205697473} a^{15} + \frac{61187399482150067169453025714735884252655547}{174224213886382681920901642340173985439882311} a^{14} + \frac{92382660090546954736763398265769588584280201}{348448427772765363841803284680347970879764622} a^{13} + \frac{6572489036655679323775871617786681578005367}{348448427772765363841803284680347970879764622} a^{12} - \frac{159036671342549139607822487715862100116899261}{348448427772765363841803284680347970879764622} a^{11} - \frac{27820326915074798722407818192682015365536157}{348448427772765363841803284680347970879764622} a^{10} + \frac{20275553621091531726610356778004638018637357}{348448427772765363841803284680347970879764622} a^{9} + \frac{161915036592372071178743419675252161961535069}{348448427772765363841803284680347970879764622} a^{8} + \frac{186904222000569607758057959962401017062082483}{696896855545530727683606569360695941759529244} a^{7} - \frac{43223820774293883280611490890067478951551185}{99556693649361532526229509908670848822789892} a^{6} + \frac{103015772266849817793781436431037637187674351}{696896855545530727683606569360695941759529244} a^{5} - \frac{42892379387213350317679855048545454498284607}{696896855545530727683606569360695941759529244} a^{4} + \frac{110907218715539591543447560807797538640196337}{696896855545530727683606569360695941759529244} a^{3} - \frac{305919527770924389541687364503581502837538413}{696896855545530727683606569360695941759529244} a^{2} - \frac{12121892493127137724365272386769566594356611}{696896855545530727683606569360695941759529244} a - \frac{30718472271020744046307715835873428240212505}{696896855545530727683606569360695941759529244}$

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:  \( 23810938178.7 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T925:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 409600
The 190 conjugacy class representatives for t20n925 are not computed
Character table for t20n925 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 10.10.728703488000000.1

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 R ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ $20$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{5}$

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
5Data not computed
97Data not computed
1193Data not computed