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Properties

Label 20.2.93209492851...3696.1
Degree $20$
Signature $[2, 9]$
Discriminant $-\,2^{44}\cdot 2657^{4}\cdot 10631$
Root discriminant $35.36$
Ramified primes $2, 2657, 10631$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1036

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-178, -632, 696, -728, 1288, -1940, 924, -1984, 610, -1504, 394, -720, 303, -224, 152, -40, 52, -4, 10, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 10*x^18 - 4*x^17 + 52*x^16 - 40*x^15 + 152*x^14 - 224*x^13 + 303*x^12 - 720*x^11 + 394*x^10 - 1504*x^9 + 610*x^8 - 1984*x^7 + 924*x^6 - 1940*x^5 + 1288*x^4 - 728*x^3 + 696*x^2 - 632*x - 178)
 
gp: K = bnfinit(x^20 + 10*x^18 - 4*x^17 + 52*x^16 - 40*x^15 + 152*x^14 - 224*x^13 + 303*x^12 - 720*x^11 + 394*x^10 - 1504*x^9 + 610*x^8 - 1984*x^7 + 924*x^6 - 1940*x^5 + 1288*x^4 - 728*x^3 + 696*x^2 - 632*x - 178, 1)
 

Normalized defining polynomial

\( x^{20} + 10 x^{18} - 4 x^{17} + 52 x^{16} - 40 x^{15} + 152 x^{14} - 224 x^{13} + 303 x^{12} - 720 x^{11} + 394 x^{10} - 1504 x^{9} + 610 x^{8} - 1984 x^{7} + 924 x^{6} - 1940 x^{5} + 1288 x^{4} - 728 x^{3} + 696 x^{2} - 632 x - 178 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-9320949285183413497763458973696=-\,2^{44}\cdot 2657^{4}\cdot 10631\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $35.36$
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:  $2, 2657, 10631$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{12}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{13}$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{14}$, $\frac{1}{245194298753421063486264022} a^{19} - \frac{20951402986049327677391061}{245194298753421063486264022} a^{18} - \frac{12154420797290228546517107}{245194298753421063486264022} a^{17} - \frac{30322985762062582270146660}{122597149376710531743132011} a^{16} + \frac{26315278878668404868659377}{245194298753421063486264022} a^{15} - \frac{13935885947737802435305131}{245194298753421063486264022} a^{14} + \frac{2390781726983365395527743}{22290390795765551226024002} a^{13} - \frac{58839555070241400344115998}{122597149376710531743132011} a^{12} - \frac{31452111616285153522071218}{122597149376710531743132011} a^{11} - \frac{8843730722062708537603658}{122597149376710531743132011} a^{10} - \frac{52122972598099939589942236}{122597149376710531743132011} a^{9} + \frac{15655176098290144519841534}{122597149376710531743132011} a^{8} + \frac{5083812985750807298542406}{11145195397882775613012001} a^{7} + \frac{414115993268378856632219}{122597149376710531743132011} a^{6} - \frac{33170841673116892949763525}{122597149376710531743132011} a^{5} + \frac{36024941276841409878083376}{122597149376710531743132011} a^{4} + \frac{30360957767862905962938279}{122597149376710531743132011} a^{3} - \frac{58972470506011055596096379}{122597149376710531743132011} a^{2} - \frac{49472940460942709316262427}{122597149376710531743132011} a - \frac{2016489003663694000855867}{122597149376710531743132011}$

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:  $10$
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:  \( 37777564.0629 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1036:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 14745600
The 396 conjugacy class representatives for t20n1036 are not computed
Character table for t20n1036 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 10.6.925322313728.1

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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.12.0.1}{12} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$2$2.8.16.13$x^{8} + 6 x^{6} + 4 x^{5} + 2 x^{4} + 4$$4$$2$$16$$D_4\times C_2$$[2, 2, 3]^{2}$
2.12.28.225$x^{12} + 4 x^{11} + 2 x^{10} + 2 x^{8} + 4 x^{7} + 4 x^{6} + 4 x^{5} + 4 x^{2} - 2$$12$$1$$28$$C_2 \times S_4$$[8/3, 8/3, 3]_{3}^{2}$
2657Data not computed
10631Data not computed