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Properties

Label 20.14.1640505640...2992.1
Degree $20$
Signature $[14, 3]$
Discriminant $-\,2^{20}\cdot 11^{16}\cdot 23^{7}$
Root discriminant $40.81$
Ramified primes $2, 11, 23$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T749

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -4, -21, 68, 147, -308, -177, 556, -511, -174, 869, -174, -511, 556, -177, -308, 147, 68, -21, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 - 21*x^18 + 68*x^17 + 147*x^16 - 308*x^15 - 177*x^14 + 556*x^13 - 511*x^12 - 174*x^11 + 869*x^10 - 174*x^9 - 511*x^8 + 556*x^7 - 177*x^6 - 308*x^5 + 147*x^4 + 68*x^3 - 21*x^2 - 4*x + 1)
 
gp: K = bnfinit(x^20 - 4*x^19 - 21*x^18 + 68*x^17 + 147*x^16 - 308*x^15 - 177*x^14 + 556*x^13 - 511*x^12 - 174*x^11 + 869*x^10 - 174*x^9 - 511*x^8 + 556*x^7 - 177*x^6 - 308*x^5 + 147*x^4 + 68*x^3 - 21*x^2 - 4*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} - 21 x^{18} + 68 x^{17} + 147 x^{16} - 308 x^{15} - 177 x^{14} + 556 x^{13} - 511 x^{12} - 174 x^{11} + 869 x^{10} - 174 x^{9} - 511 x^{8} + 556 x^{7} - 177 x^{6} - 308 x^{5} + 147 x^{4} + 68 x^{3} - 21 x^{2} - 4 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[14, 3]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-164050564045619941441358756052992=-\,2^{20}\cdot 11^{16}\cdot 23^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $40.81$
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:  $2, 11, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{10739334409} a^{18} - \frac{3735079095}{10739334409} a^{17} - \frac{528075318}{10739334409} a^{16} - \frac{277913483}{10739334409} a^{15} + \frac{1031280938}{10739334409} a^{14} - \frac{5240838875}{10739334409} a^{13} + \frac{1626535809}{10739334409} a^{12} + \frac{31977280}{10739334409} a^{11} - \frac{2821879165}{10739334409} a^{10} - \frac{2889837763}{10739334409} a^{9} - \frac{2821879165}{10739334409} a^{8} + \frac{31977280}{10739334409} a^{7} + \frac{1626535809}{10739334409} a^{6} - \frac{5240838875}{10739334409} a^{5} + \frac{1031280938}{10739334409} a^{4} - \frac{277913483}{10739334409} a^{3} - \frac{528075318}{10739334409} a^{2} - \frac{3735079095}{10739334409} a + \frac{1}{10739334409}$, $\frac{1}{10739334409} a^{19} - \frac{182651391}{466927583} a^{17} + \frac{1622777891}{10739334409} a^{16} - \frac{583578467}{10739334409} a^{15} + \frac{4403036927}{10739334409} a^{14} - \frac{515967797}{10739334409} a^{13} + \frac{1007648258}{10739334409} a^{12} - \frac{1498627200}{10739334409} a^{11} - \frac{25224335}{466927583} a^{10} + \frac{2001863391}{10739334409} a^{9} + \frac{2341655338}{10739334409} a^{8} + \frac{2949787774}{10739334409} a^{7} - \frac{4265167897}{10739334409} a^{6} - \frac{1111222668}{10739334409} a^{5} - \frac{59711830}{466927583} a^{4} - \frac{2142934723}{10739334409} a^{3} - \frac{1834387721}{10739334409} a^{2} - \frac{3672906674}{10739334409} a + \frac{3735079095}{10739334409}$

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:  $16$
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:  \( 1558002565.37 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T749:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 81920
The 332 conjugacy class representatives for t20n749 are not computed
Character table for t20n749 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.10.116117348402176.2

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 R ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/5.10.0.1}{10} }{,}\,{\href{/LocalNumberField/5.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }^{2}$ R ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
2Data not computed
$11$11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
$23$$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.4.3.1$x^{4} + 46$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
23.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
23.4.3.2$x^{4} - 23$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$