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Properties

Label 20.4.53990223619...3125.1
Degree $20$
Signature $[4, 8]$
Discriminant $5^{10}\cdot 11^{4}\cdot 131^{4}\cdot 128221$
Root discriminant $17.24$
Ramified primes $5, 11, 131, 128221$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T887

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 14, -82, 293, -714, 1306, -1914, 2273, -2120, 1395, -346, -642, 1262, -1400, 1155, -748, 383, -152, 45, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 9*x^19 + 45*x^18 - 152*x^17 + 383*x^16 - 748*x^15 + 1155*x^14 - 1400*x^13 + 1262*x^12 - 642*x^11 - 346*x^10 + 1395*x^9 - 2120*x^8 + 2273*x^7 - 1914*x^6 + 1306*x^5 - 714*x^4 + 293*x^3 - 82*x^2 + 14*x - 1)
 
gp: K = bnfinit(x^20 - 9*x^19 + 45*x^18 - 152*x^17 + 383*x^16 - 748*x^15 + 1155*x^14 - 1400*x^13 + 1262*x^12 - 642*x^11 - 346*x^10 + 1395*x^9 - 2120*x^8 + 2273*x^7 - 1914*x^6 + 1306*x^5 - 714*x^4 + 293*x^3 - 82*x^2 + 14*x - 1, 1)
 

Normalized defining polynomial

\( x^{20} - 9 x^{19} + 45 x^{18} - 152 x^{17} + 383 x^{16} - 748 x^{15} + 1155 x^{14} - 1400 x^{13} + 1262 x^{12} - 642 x^{11} - 346 x^{10} + 1395 x^{9} - 2120 x^{8} + 2273 x^{7} - 1914 x^{6} + 1306 x^{5} - 714 x^{4} + 293 x^{3} - 82 x^{2} + 14 x - 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(5399022361905183408203125=5^{10}\cdot 11^{4}\cdot 131^{4}\cdot 128221\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $17.24$
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:  $5, 11, 131, 128221$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{5} a^{18} + \frac{1}{5} a^{16} + \frac{2}{5} a^{15} + \frac{2}{5} a^{14} + \frac{2}{5} a^{13} + \frac{2}{5} a^{11} - \frac{1}{5} a^{8} + \frac{1}{5} a^{7} - \frac{2}{5} a^{6} + \frac{1}{5} a^{5} - \frac{2}{5} a^{4} - \frac{1}{5} a^{3} + \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{1025149968175} a^{19} - \frac{12750394542}{1025149968175} a^{18} - \frac{477308810789}{1025149968175} a^{17} - \frac{8705685253}{205029993635} a^{16} + \frac{319134968033}{1025149968175} a^{15} + \frac{247527426498}{1025149968175} a^{14} - \frac{355540803369}{1025149968175} a^{13} + \frac{359801814237}{1025149968175} a^{12} - \frac{412192710184}{1025149968175} a^{11} - \frac{75526553297}{205029993635} a^{10} - \frac{228664378266}{1025149968175} a^{9} - \frac{56277649177}{1025149968175} a^{8} - \frac{212554304359}{1025149968175} a^{7} - \frac{14604685391}{41005998727} a^{6} + \frac{398714124026}{1025149968175} a^{5} - \frac{474135812797}{1025149968175} a^{4} + \frac{87118776027}{1025149968175} a^{3} - \frac{63453942678}{1025149968175} a^{2} + \frac{471505120842}{1025149968175} a - \frac{319832657437}{1025149968175}$

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:  \( 24644.9655838 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T887:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 245760
The 201 conjugacy class representatives for t20n887 are not computed
Character table for t20n887 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.1.36025.1, 10.2.6489003125.1

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 $20$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{3}$ R ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ $20$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$11$11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.6.0.1$x^{6} + x^{2} - 2 x + 8$$1$$6$$0$$C_6$$[\ ]^{6}$
11.6.0.1$x^{6} + x^{2} - 2 x + 8$$1$$6$$0$$C_6$$[\ ]^{6}$
$131$$\Q_{131}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{131}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{131}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{131}$$x + 3$$1$$1$$0$Trivial$[\ ]$
131.2.0.1$x^{2} - x + 14$$1$$2$$0$$C_2$$[\ ]^{2}$
131.2.1.2$x^{2} + 393$$2$$1$$1$$C_2$$[\ ]_{2}$
131.2.1.2$x^{2} + 393$$2$$1$$1$$C_2$$[\ ]_{2}$
131.2.0.1$x^{2} - x + 14$$1$$2$$0$$C_2$$[\ ]^{2}$
131.4.2.1$x^{4} + 3537 x^{2} + 3363556$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
131.4.0.1$x^{4} - x + 6$$1$$4$$0$$C_4$$[\ ]^{4}$
128221Data not computed