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Properties

Label 20.6.10929603344...9968.4
Degree $20$
Signature $[6, 7]$
Discriminant $-\,2^{40}\cdot 1583^{5}$
Root discriminant $25.23$
Ramified primes $2, 1583$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T756

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![17, -36, -292, 1136, -1575, 1292, -816, -296, 1237, -176, -2132, 2932, -1623, 268, 52, 0, -14, 12, 0, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 12*x^17 - 14*x^16 + 52*x^14 + 268*x^13 - 1623*x^12 + 2932*x^11 - 2132*x^10 - 176*x^9 + 1237*x^8 - 296*x^7 - 816*x^6 + 1292*x^5 - 1575*x^4 + 1136*x^3 - 292*x^2 - 36*x + 17)
 
gp: K = bnfinit(x^20 - 4*x^19 + 12*x^17 - 14*x^16 + 52*x^14 + 268*x^13 - 1623*x^12 + 2932*x^11 - 2132*x^10 - 176*x^9 + 1237*x^8 - 296*x^7 - 816*x^6 + 1292*x^5 - 1575*x^4 + 1136*x^3 - 292*x^2 - 36*x + 17, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 12 x^{17} - 14 x^{16} + 52 x^{14} + 268 x^{13} - 1623 x^{12} + 2932 x^{11} - 2132 x^{10} - 176 x^{9} + 1237 x^{8} - 296 x^{7} - 816 x^{6} + 1292 x^{5} - 1575 x^{4} + 1136 x^{3} - 292 x^{2} - 36 x + 17 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-10929603344110461118702419968=-\,2^{40}\cdot 1583^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $25.23$
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:  $2, 1583$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{89} a^{18} - \frac{35}{89} a^{17} - \frac{30}{89} a^{16} + \frac{6}{89} a^{15} - \frac{36}{89} a^{14} + \frac{33}{89} a^{13} + \frac{9}{89} a^{12} + \frac{40}{89} a^{11} + \frac{7}{89} a^{10} + \frac{34}{89} a^{9} - \frac{44}{89} a^{8} + \frac{35}{89} a^{7} - \frac{5}{89} a^{6} - \frac{6}{89} a^{5} - \frac{39}{89} a^{4} + \frac{24}{89} a^{3} - \frac{41}{89} a^{2} + \frac{33}{89} a - \frac{11}{89}$, $\frac{1}{5252805415218916645938340205705} a^{19} + \frac{6121305990133013275223041073}{5252805415218916645938340205705} a^{18} + \frac{671615396207418561224562524566}{5252805415218916645938340205705} a^{17} - \frac{2080132656184157013456419105661}{5252805415218916645938340205705} a^{16} + \frac{2164994665162143855889845674729}{5252805415218916645938340205705} a^{15} - \frac{171469665907442130066689302912}{5252805415218916645938340205705} a^{14} - \frac{1337176780715625394925341525537}{5252805415218916645938340205705} a^{13} + \frac{1567646802653845395539113047094}{5252805415218916645938340205705} a^{12} + \frac{343149763935792843218054312931}{1050561083043783329187668041141} a^{11} + \frac{1897625210385323146401280027427}{5252805415218916645938340205705} a^{10} - \frac{2418803176889412862181403477653}{5252805415218916645938340205705} a^{9} - \frac{2031502387474094376212079854092}{5252805415218916645938340205705} a^{8} + \frac{564358265668170001765047134593}{5252805415218916645938340205705} a^{7} + \frac{396181305857014231045156159640}{1050561083043783329187668041141} a^{6} - \frac{2446162346362036843204336550416}{5252805415218916645938340205705} a^{5} - \frac{86135968075619616497091108986}{1050561083043783329187668041141} a^{4} + \frac{97601028266529877998174313062}{1050561083043783329187668041141} a^{3} - \frac{2118214208934477790146695426314}{5252805415218916645938340205705} a^{2} - \frac{376688260263839122093416312895}{1050561083043783329187668041141} a - \frac{886559915719298485645586622151}{5252805415218916645938340205705}$

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:  $12$
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:  \( 1784207.36364 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T756:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 102400
The 130 conjugacy class representatives for t20n756 are not computed
Character table for t20n756 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 10.6.82112970752.1

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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }$ ${\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{5}$

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
1583Data not computed