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Properties

Label 20.8.46179397182...4576.1
Degree $20$
Signature $[8, 6]$
Discriminant $2^{20}\cdot 11^{18}\cdot 89^{2}$
Root discriminant $27.12$
Ramified primes $2, 11, 89$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T340

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-43, -158, 604, 952, -2713, 4392, -2555, -3350, 8897, -11352, 10330, -7568, 4823, -2726, 1403, -652, 251, -84, 29, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 29*x^18 - 84*x^17 + 251*x^16 - 652*x^15 + 1403*x^14 - 2726*x^13 + 4823*x^12 - 7568*x^11 + 10330*x^10 - 11352*x^9 + 8897*x^8 - 3350*x^7 - 2555*x^6 + 4392*x^5 - 2713*x^4 + 952*x^3 + 604*x^2 - 158*x - 43)
 
gp: K = bnfinit(x^20 - 8*x^19 + 29*x^18 - 84*x^17 + 251*x^16 - 652*x^15 + 1403*x^14 - 2726*x^13 + 4823*x^12 - 7568*x^11 + 10330*x^10 - 11352*x^9 + 8897*x^8 - 3350*x^7 - 2555*x^6 + 4392*x^5 - 2713*x^4 + 952*x^3 + 604*x^2 - 158*x - 43, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 29 x^{18} - 84 x^{17} + 251 x^{16} - 652 x^{15} + 1403 x^{14} - 2726 x^{13} + 4823 x^{12} - 7568 x^{11} + 10330 x^{10} - 11352 x^{9} + 8897 x^{8} - 3350 x^{7} - 2555 x^{6} + 4392 x^{5} - 2713 x^{4} + 952 x^{3} + 604 x^{2} - 158 x - 43 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(46179397182603358960092184576=2^{20}\cdot 11^{18}\cdot 89^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.12$
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:  $2, 11, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{11} a^{15} + \frac{2}{11} a^{14} - \frac{2}{11} a^{13} - \frac{5}{11} a^{12} - \frac{2}{11} a^{11} - \frac{1}{11} a^{10} - \frac{1}{11} a^{9} - \frac{1}{11} a^{8} - \frac{1}{11} a^{7} - \frac{1}{11} a^{6} - \frac{1}{11} a^{5} - \frac{2}{11} a^{4} - \frac{3}{11} a^{3} + \frac{1}{11} a^{2} + \frac{4}{11} a + \frac{1}{11}$, $\frac{1}{11} a^{16} + \frac{5}{11} a^{14} - \frac{1}{11} a^{13} - \frac{3}{11} a^{12} + \frac{3}{11} a^{11} + \frac{1}{11} a^{10} + \frac{1}{11} a^{9} + \frac{1}{11} a^{8} + \frac{1}{11} a^{7} + \frac{1}{11} a^{6} + \frac{1}{11} a^{4} - \frac{4}{11} a^{3} + \frac{2}{11} a^{2} + \frac{4}{11} a - \frac{2}{11}$, $\frac{1}{11} a^{17} - \frac{4}{11} a^{13} - \frac{5}{11} a^{12} - \frac{5}{11} a^{10} - \frac{5}{11} a^{9} - \frac{5}{11} a^{8} - \frac{5}{11} a^{7} + \frac{5}{11} a^{6} - \frac{5}{11} a^{5} - \frac{5}{11} a^{4} - \frac{5}{11} a^{3} - \frac{1}{11} a^{2} - \frac{5}{11}$, $\frac{1}{11} a^{18} - \frac{4}{11} a^{14} - \frac{5}{11} a^{13} - \frac{5}{11} a^{11} - \frac{5}{11} a^{10} - \frac{5}{11} a^{9} - \frac{5}{11} a^{8} + \frac{5}{11} a^{7} - \frac{5}{11} a^{6} - \frac{5}{11} a^{5} - \frac{5}{11} a^{4} - \frac{1}{11} a^{3} - \frac{5}{11} a$, $\frac{1}{92577783077169057244392937} a^{19} + \frac{3084196827872139562178054}{92577783077169057244392937} a^{18} - \frac{820294157591080915399963}{92577783077169057244392937} a^{17} + \frac{1540494529700229484193103}{92577783077169057244392937} a^{16} - \frac{442856056006839851404140}{92577783077169057244392937} a^{15} + \frac{6435544536441644986103657}{92577783077169057244392937} a^{14} - \frac{26080032802091892938686825}{92577783077169057244392937} a^{13} - \frac{19330454493719528773396284}{92577783077169057244392937} a^{12} - \frac{18815503090783844329282322}{92577783077169057244392937} a^{11} + \frac{29178475866790997216234757}{92577783077169057244392937} a^{10} - \frac{4042910055908032499357626}{92577783077169057244392937} a^{9} - \frac{11696986922650753781380530}{92577783077169057244392937} a^{8} + \frac{4292033059823087858001872}{92577783077169057244392937} a^{7} + \frac{34151553782171669983272970}{92577783077169057244392937} a^{6} + \frac{40720949463077447589637203}{92577783077169057244392937} a^{5} - \frac{11265719780432720812271376}{92577783077169057244392937} a^{4} + \frac{748143488957798377605930}{8416162097924459749490267} a^{3} - \frac{35714297954588535395208475}{92577783077169057244392937} a^{2} + \frac{9063950284263843574705604}{92577783077169057244392937} a + \frac{856546980772614252901072}{2152971699469047842892859}$

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:  $13$
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:  \( 5522109.21263 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T340:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 5120
The 80 conjugacy class representatives for t20n340 are not computed
Character table for t20n340 is not computed

Intermediate fields

\(\Q(\sqrt{11}) \), \(\Q(\zeta_{11})^+\), \(\Q(\zeta_{44})^+\)

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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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$2.10.10.11$x^{10} - x^{8} + 3 x^{6} + x^{2} - 3$$2$$5$$10$$C_{10}$$[2]^{5}$
2.10.10.11$x^{10} - x^{8} + 3 x^{6} + x^{2} - 3$$2$$5$$10$$C_{10}$$[2]^{5}$
$11$11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$
89Data not computed