↑

Properties

Label 20.0.16105352050...4144.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{16}\cdot 3^{22}\cdot 23^{8}$
Root discriminant $20.43$
Ramified primes $2, 3, 23$
Class number $2$
Class group $[2]$
Galois group $S_5$ (as 20T32)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9, -15, 16, 21, -78, 127, -150, 69, 147, -312, 456, -454, 411, -297, 222, -117, 66, -23, 12, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 + 12*x^18 - 23*x^17 + 66*x^16 - 117*x^15 + 222*x^14 - 297*x^13 + 411*x^12 - 454*x^11 + 456*x^10 - 312*x^9 + 147*x^8 + 69*x^7 - 150*x^6 + 127*x^5 - 78*x^4 + 21*x^3 + 16*x^2 - 15*x + 9)
 
gp: K = bnfinit(x^20 - 3*x^19 + 12*x^18 - 23*x^17 + 66*x^16 - 117*x^15 + 222*x^14 - 297*x^13 + 411*x^12 - 454*x^11 + 456*x^10 - 312*x^9 + 147*x^8 + 69*x^7 - 150*x^6 + 127*x^5 - 78*x^4 + 21*x^3 + 16*x^2 - 15*x + 9, 1)
 

Normalized defining polynomial

\( x^{20} - 3 x^{19} + 12 x^{18} - 23 x^{17} + 66 x^{16} - 117 x^{15} + 222 x^{14} - 297 x^{13} + 411 x^{12} - 454 x^{11} + 456 x^{10} - 312 x^{9} + 147 x^{8} + 69 x^{7} - 150 x^{6} + 127 x^{5} - 78 x^{4} + 21 x^{3} + 16 x^{2} - 15 x + 9 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(161053520503936294265094144=2^{16}\cdot 3^{22}\cdot 23^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.43$
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:  $2, 3, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{7} + \frac{1}{3} a^{4} - \frac{1}{3} a$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{8} + \frac{1}{3} a^{5} - \frac{1}{3} a^{2}$, $\frac{1}{9} a^{12} + \frac{1}{9} a^{11} - \frac{1}{9} a^{10} + \frac{1}{9} a^{9} - \frac{2}{9} a^{8} + \frac{4}{9} a^{6} + \frac{4}{9} a^{5} - \frac{1}{9} a^{4} - \frac{2}{9} a^{3} + \frac{4}{9} a^{2} + \frac{1}{3} a$, $\frac{1}{9} a^{13} + \frac{1}{9} a^{11} - \frac{1}{9} a^{10} + \frac{2}{9} a^{8} + \frac{1}{9} a^{7} - \frac{2}{9} a^{5} - \frac{4}{9} a^{4} - \frac{1}{9} a^{2} + \frac{1}{3} a$, $\frac{1}{9} a^{14} + \frac{1}{9} a^{11} + \frac{1}{9} a^{10} + \frac{1}{9} a^{9} + \frac{1}{3} a^{6} + \frac{4}{9} a^{5} + \frac{1}{9} a^{4} + \frac{1}{9} a^{3} - \frac{4}{9} a^{2} - \frac{1}{3} a$, $\frac{1}{18} a^{15} - \frac{1}{18} a^{12} + \frac{1}{9} a^{11} - \frac{1}{9} a^{9} - \frac{4}{9} a^{8} + \frac{1}{3} a^{7} - \frac{2}{9} a^{6} - \frac{2}{9} a^{5} - \frac{1}{2} a^{3} + \frac{2}{9} a^{2} + \frac{1}{3} a - \frac{1}{2}$, $\frac{1}{18} a^{16} - \frac{1}{18} a^{13} - \frac{1}{9} a^{11} + \frac{1}{9} a^{9} + \frac{2}{9} a^{8} + \frac{4}{9} a^{7} + \frac{1}{3} a^{6} - \frac{4}{9} a^{5} - \frac{7}{18} a^{4} + \frac{1}{9} a^{3} - \frac{4}{9} a^{2} - \frac{1}{6} a$, $\frac{1}{18} a^{17} - \frac{1}{18} a^{14} + \frac{1}{9} a^{11} - \frac{1}{9} a^{8} + \frac{1}{18} a^{5} - \frac{1}{18} a^{2}$, $\frac{1}{162} a^{18} + \frac{1}{81} a^{17} - \frac{1}{54} a^{16} + \frac{1}{81} a^{15} - \frac{1}{81} a^{14} + \frac{1}{54} a^{13} - \frac{1}{162} a^{12} - \frac{4}{81} a^{11} - \frac{1}{9} a^{10} + \frac{8}{81} a^{9} - \frac{2}{81} a^{8} + \frac{1}{27} a^{7} + \frac{79}{162} a^{6} - \frac{23}{81} a^{5} - \frac{5}{54} a^{4} + \frac{7}{81} a^{3} - \frac{19}{81} a^{2} - \frac{5}{54} a - \frac{5}{18}$, $\frac{1}{160749696474} a^{19} + \frac{20075465}{26791616079} a^{18} - \frac{1824513287}{80374848237} a^{17} - \frac{2406521503}{160749696474} a^{16} - \frac{219269486}{8930538693} a^{15} - \frac{2660712952}{80374848237} a^{14} + \frac{2118915334}{80374848237} a^{13} + \frac{86657285}{8930538693} a^{12} - \frac{1115018149}{80374848237} a^{11} + \frac{6778201823}{80374848237} a^{10} - \frac{821106472}{26791616079} a^{9} - \frac{7782749927}{80374848237} a^{8} - \frac{44605974011}{160749696474} a^{7} + \frac{6294788179}{26791616079} a^{6} + \frac{22498882786}{80374848237} a^{5} + \frac{41567703659}{160749696474} a^{4} + \frac{7642784165}{26791616079} a^{3} - \frac{11592075811}{80374848237} a^{2} - \frac{2680035904}{26791616079} a - \frac{1691606518}{8930538693}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}$, which has order $2$

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $9$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{232294184}{2976846231} a^{19} + \frac{4375258585}{17861077386} a^{18} - \frac{1864489903}{1984564154} a^{17} + \frac{16716790891}{8930538693} a^{16} - \frac{45999889924}{8930538693} a^{15} + \frac{171213886615}{17861077386} a^{14} - \frac{155791007006}{8930538693} a^{13} + \frac{436345370111}{17861077386} a^{12} - \frac{293119401211}{8930538693} a^{11} + \frac{346710509297}{8930538693} a^{10} - \frac{345828051655}{8930538693} a^{9} + \frac{270456812641}{8930538693} a^{8} - \frac{156481297835}{8930538693} a^{7} + \frac{60526036675}{17861077386} a^{6} + \frac{93045477361}{17861077386} a^{5} - \frac{27530735567}{8930538693} a^{4} + \frac{11474653574}{2976846231} a^{3} - \frac{36576409739}{17861077386} a^{2} + \frac{122165722}{2976846231} a + \frac{1047062401}{1984564154} \) (order $6$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 314083.885561 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_5$ (as 20T32):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 120
The 7 conjugacy class representatives for $S_5$
Character table for $S_5$

Intermediate fields

\(\Q(\sqrt{-3}) \), 10.4.1410076263168.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 5 sibling: data not computed
Degree 6 sibling: data not computed
Degree 10 siblings: data not computed
Degree 12 sibling: data not computed
Degree 15 sibling: data not computed
Degree 20 siblings: data not computed
Degree 24 sibling: data not computed
Degree 30 siblings: data not computed
Degree 40 sibling: 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 R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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
$2$2.10.8.1$x^{10} - 2 x^{5} + 4$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
2.10.8.1$x^{10} - 2 x^{5} + 4$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
$3$3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.6.7.5$x^{6} + 6 x^{2} + 3$$6$$1$$7$$D_{6}$$[3/2]_{2}^{2}$
3.12.14.11$x^{12} + 6 x^{11} + 21 x^{10} + 36 x^{9} + 30 x^{8} + 36 x^{7} + 3 x^{6} + 36 x^{5} + 27 x^{4} - 9 x^{2} + 36$$6$$2$$14$$D_6$$[3/2]_{2}^{2}$
$23$23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.4.2.2$x^{4} - 23 x^{2} + 3703$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
23.4.2.2$x^{4} - 23 x^{2} + 3703$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
23.4.2.2$x^{4} - 23 x^{2} + 3703$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
23.4.2.2$x^{4} - 23 x^{2} + 3703$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$