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Properties

Label 20.4.34843561790...8125.1
Degree $20$
Signature $[4, 8]$
Discriminant $5^{15}\cdot 97^{2}\cdot 3319^{4}$
Root discriminant $26.74$
Ramified primes $5, 97, 3319$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T755

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-3089, 10968, -20808, 27828, -28282, 22909, -13689, 5123, -346, -1132, 465, 559, -715, 238, 109, -71, -28, 27, -2, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 - 2*x^18 + 27*x^17 - 28*x^16 - 71*x^15 + 109*x^14 + 238*x^13 - 715*x^12 + 559*x^11 + 465*x^10 - 1132*x^9 - 346*x^8 + 5123*x^7 - 13689*x^6 + 22909*x^5 - 28282*x^4 + 27828*x^3 - 20808*x^2 + 10968*x - 3089)
 
gp: K = bnfinit(x^20 - 4*x^19 - 2*x^18 + 27*x^17 - 28*x^16 - 71*x^15 + 109*x^14 + 238*x^13 - 715*x^12 + 559*x^11 + 465*x^10 - 1132*x^9 - 346*x^8 + 5123*x^7 - 13689*x^6 + 22909*x^5 - 28282*x^4 + 27828*x^3 - 20808*x^2 + 10968*x - 3089, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} - 2 x^{18} + 27 x^{17} - 28 x^{16} - 71 x^{15} + 109 x^{14} + 238 x^{13} - 715 x^{12} + 559 x^{11} + 465 x^{10} - 1132 x^{9} - 346 x^{8} + 5123 x^{7} - 13689 x^{6} + 22909 x^{5} - 28282 x^{4} + 27828 x^{3} - 20808 x^{2} + 10968 x - 3089 \)

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:  \(34843561790753768585205078125=5^{15}\cdot 97^{2}\cdot 3319^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.74$
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:  $5, 97, 3319$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{19} a^{18} + \frac{6}{19} a^{17} - \frac{6}{19} a^{16} + \frac{1}{19} a^{15} + \frac{5}{19} a^{14} - \frac{9}{19} a^{13} + \frac{3}{19} a^{12} + \frac{8}{19} a^{11} + \frac{9}{19} a^{10} + \frac{4}{19} a^{9} + \frac{5}{19} a^{8} - \frac{8}{19} a^{7} - \frac{5}{19} a^{6} - \frac{1}{19} a^{5} - \frac{3}{19} a^{4} - \frac{9}{19} a^{3} - \frac{3}{19} a^{2} + \frac{7}{19} a - \frac{7}{19}$, $\frac{1}{37333124133188552825414138859375341339} a^{19} - \frac{899366262252785022354496068331170905}{37333124133188552825414138859375341339} a^{18} - \frac{16862901001109150637822272360329070412}{37333124133188552825414138859375341339} a^{17} + \frac{16813255895099013620924335087703116004}{37333124133188552825414138859375341339} a^{16} + \frac{5240027102123721455859821180900289190}{37333124133188552825414138859375341339} a^{15} - \frac{1126697854680343445040745535690695819}{37333124133188552825414138859375341339} a^{14} - \frac{8921663542646645298075026606032169857}{37333124133188552825414138859375341339} a^{13} + \frac{18095820118131559767170369565562718132}{37333124133188552825414138859375341339} a^{12} - \frac{2169803925483821629310771100933888725}{37333124133188552825414138859375341339} a^{11} - \frac{277863198790076855088942374960097993}{1964901270167818569758638887335544281} a^{10} + \frac{17791056376990000222601474350997004823}{37333124133188552825414138859375341339} a^{9} + \frac{3276200617216469209847801304897367950}{37333124133188552825414138859375341339} a^{8} + \frac{2590688185657785773451853710381655023}{37333124133188552825414138859375341339} a^{7} - \frac{7269797252133331945965531012879556171}{37333124133188552825414138859375341339} a^{6} - \frac{5666689944220456260061280946846206622}{37333124133188552825414138859375341339} a^{5} - \frac{15865892915161879368943245595424617615}{37333124133188552825414138859375341339} a^{4} + \frac{2070628234149566192317405240333309733}{37333124133188552825414138859375341339} a^{3} - \frac{1080386032991580845748562074528987588}{37333124133188552825414138859375341339} a^{2} + \frac{8992569882169759348722241506918677751}{37333124133188552825414138859375341339} a - \frac{9785141773364272946715887001281475948}{37333124133188552825414138859375341339}$

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:  \( 2315293.58671 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T755:

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 t20n755 are not computed
Character table for t20n755 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.6.34424253125.1

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 $20$ ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ $20$ $20$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ $20$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}$ $20$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ $20$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}$

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
$97$97.4.2.2$x^{4} - 97 x^{2} + 47045$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
97.8.0.1$x^{8} - x + 84$$1$$8$$0$$C_8$$[\ ]^{8}$
97.8.0.1$x^{8} - x + 84$$1$$8$$0$$C_8$$[\ ]^{8}$
3319Data not computed