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Properties

Label 20.8.82270188117...0144.5
Degree $20$
Signature $[8, 6]$
Discriminant $2^{40}\cdot 11^{17}\cdot 23^{6}$
Root discriminant $78.66$
Ramified primes $2, 11, 23$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T326

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5819, 0, -186208, 0, -1468159, 0, -1875236, 0, -360173, 0, 346116, 0, 74625, 0, -9024, 0, -520, 0, 36, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 36*x^18 - 520*x^16 - 9024*x^14 + 74625*x^12 + 346116*x^10 - 360173*x^8 - 1875236*x^6 - 1468159*x^4 - 186208*x^2 + 5819)
 
gp: K = bnfinit(x^20 + 36*x^18 - 520*x^16 - 9024*x^14 + 74625*x^12 + 346116*x^10 - 360173*x^8 - 1875236*x^6 - 1468159*x^4 - 186208*x^2 + 5819, 1)
 

Normalized defining polynomial

\( x^{20} + 36 x^{18} - 520 x^{16} - 9024 x^{14} + 74625 x^{12} + 346116 x^{10} - 360173 x^{8} - 1875236 x^{6} - 1468159 x^{4} - 186208 x^{2} + 5819 \)

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:  \(82270188117030423168911138645967110144=2^{40}\cdot 11^{17}\cdot 23^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $78.66$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{23} a^{16} - \frac{10}{23} a^{14} + \frac{9}{23} a^{12} - \frac{8}{23} a^{10} - \frac{10}{23} a^{8} - \frac{11}{23} a^{6} + \frac{7}{23} a^{4}$, $\frac{1}{23} a^{17} - \frac{10}{23} a^{15} + \frac{9}{23} a^{13} - \frac{8}{23} a^{11} - \frac{10}{23} a^{9} - \frac{11}{23} a^{7} + \frac{7}{23} a^{5}$, $\frac{1}{68288568599395385032947262523989} a^{18} - \frac{951515607847179538855016327874}{68288568599395385032947262523989} a^{16} - \frac{20906520510575508704118370807108}{68288568599395385032947262523989} a^{14} - \frac{28826697238252785582137283887897}{68288568599395385032947262523989} a^{12} - \frac{1531873615450233191952148760991}{68288568599395385032947262523989} a^{10} - \frac{30295639418793068572888076232900}{68288568599395385032947262523989} a^{8} + \frac{3761750778977482291358656048127}{68288568599395385032947262523989} a^{6} + \frac{29323439578147136754421190938756}{68288568599395385032947262523989} a^{4} - \frac{792176398940812095754114746450}{2969068199973712392736837501043} a^{2} + \frac{546835774029621297093561482754}{2969068199973712392736837501043}$, $\frac{1}{68288568599395385032947262523989} a^{19} - \frac{951515607847179538855016327874}{68288568599395385032947262523989} a^{17} - \frac{20906520510575508704118370807108}{68288568599395385032947262523989} a^{15} - \frac{28826697238252785582137283887897}{68288568599395385032947262523989} a^{13} - \frac{1531873615450233191952148760991}{68288568599395385032947262523989} a^{11} - \frac{30295639418793068572888076232900}{68288568599395385032947262523989} a^{9} + \frac{3761750778977482291358656048127}{68288568599395385032947262523989} a^{7} + \frac{29323439578147136754421190938756}{68288568599395385032947262523989} a^{5} - \frac{792176398940812095754114746450}{2969068199973712392736837501043} a^{3} + \frac{546835774029621297093561482754}{2969068199973712392736837501043} a$

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:  \( 274841012368 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T326:

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 t20n326 are not computed
Character table for t20n326 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.10.116117348402176.1

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 ${\href{/LocalNumberField/3.10.0.1}{10} }{,}\,{\href{/LocalNumberField/3.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }{,}\,{\href{/LocalNumberField/13.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }^{2}$

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
2Data not computed
$11$11.10.9.7$x^{10} + 2673$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
$23$$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.4.3.2$x^{4} - 23$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
23.4.3.2$x^{4} - 23$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$