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Properties

Label 20.6.76601707609...5747.1
Degree $20$
Signature $[6, 7]$
Discriminant $-\,11^{16}\cdot 307\cdot 7369^{2}$
Root discriminant $22.09$
Ramified primes $11, 307, 7369$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T846

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 14, 66, -77, 39, 20, -300, 503, -439, -291, 968, -671, -62, 257, -48, -88, 35, 15, -7, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 - 7*x^18 + 15*x^17 + 35*x^16 - 88*x^15 - 48*x^14 + 257*x^13 - 62*x^12 - 671*x^11 + 968*x^10 - 291*x^9 - 439*x^8 + 503*x^7 - 300*x^6 + 20*x^5 + 39*x^4 - 77*x^3 + 66*x^2 + 14*x - 1)
 
gp: K = bnfinit(x^20 - x^19 - 7*x^18 + 15*x^17 + 35*x^16 - 88*x^15 - 48*x^14 + 257*x^13 - 62*x^12 - 671*x^11 + 968*x^10 - 291*x^9 - 439*x^8 + 503*x^7 - 300*x^6 + 20*x^5 + 39*x^4 - 77*x^3 + 66*x^2 + 14*x - 1, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} - 7 x^{18} + 15 x^{17} + 35 x^{16} - 88 x^{15} - 48 x^{14} + 257 x^{13} - 62 x^{12} - 671 x^{11} + 968 x^{10} - 291 x^{9} - 439 x^{8} + 503 x^{7} - 300 x^{6} + 20 x^{5} + 39 x^{4} - 77 x^{3} + 66 x^{2} + 14 x - 1 \)

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:  \(-766017076090168481174155747=-\,11^{16}\cdot 307\cdot 7369^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $22.09$
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:  $11, 307, 7369$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{89} a^{18} + \frac{16}{89} a^{17} + \frac{38}{89} a^{16} - \frac{34}{89} a^{15} - \frac{2}{89} a^{14} + \frac{31}{89} a^{13} + \frac{43}{89} a^{12} + \frac{3}{89} a^{11} + \frac{18}{89} a^{10} + \frac{22}{89} a^{9} + \frac{15}{89} a^{8} + \frac{43}{89} a^{7} + \frac{2}{89} a^{6} + \frac{32}{89} a^{5} - \frac{32}{89} a^{4} + \frac{44}{89} a^{3} + \frac{41}{89} a^{2} - \frac{23}{89} a - \frac{20}{89}$, $\frac{1}{162263641843910912390064409} a^{19} - \frac{807912781213877368753320}{162263641843910912390064409} a^{18} - \frac{37918927006951240772097349}{162263641843910912390064409} a^{17} - \frac{27668294685061429555606890}{162263641843910912390064409} a^{16} - \frac{52209794420841331228081173}{162263641843910912390064409} a^{15} + \frac{69530327918013691333236277}{162263641843910912390064409} a^{14} + \frac{19244477461420338651751619}{162263641843910912390064409} a^{13} + \frac{10467097109074558125838960}{162263641843910912390064409} a^{12} + \frac{66853150244422317502696032}{162263641843910912390064409} a^{11} + \frac{43125823490997644297191293}{162263641843910912390064409} a^{10} - \frac{79484188852267057744553695}{162263641843910912390064409} a^{9} - \frac{74195910707054149059918329}{162263641843910912390064409} a^{8} + \frac{45592954063903157445740238}{162263641843910912390064409} a^{7} + \frac{5260269967161917151653550}{162263641843910912390064409} a^{6} - \frac{30378441624026542809206848}{162263641843910912390064409} a^{5} - \frac{22661103943653458961927456}{162263641843910912390064409} a^{4} - \frac{13776919127830904630108140}{162263641843910912390064409} a^{3} - \frac{17680643449550668007436227}{162263641843910912390064409} a^{2} - \frac{9118178908587134786196706}{162263641843910912390064409} a - \frac{7517548171316199577564687}{162263641843910912390064409}$

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:  \( 340648.791996 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T846:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 163840
The 649 conjugacy class representatives for t20n846 are not computed
Character table for t20n846 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.10.1579610594089.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.10.0.1}{10} }{,}\,{\href{/LocalNumberField/2.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/3.10.0.1}{10} }{,}\,{\href{/LocalNumberField/3.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/5.10.0.1}{10} }{,}\,{\href{/LocalNumberField/5.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R $20$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ $20$ $20$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }^{2}$

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
$11$11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
307Data not computed
7369Data not computed