↑

Properties

Label 20.6.69654621282...6096.2
Degree $20$
Signature $[6, 7]$
Discriminant $-\,2^{10}\cdot 11^{16}\cdot 23^{6}$
Root discriminant $24.67$
Ramified primes $2, 11, 23$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T326

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-32, 176, -368, 652, -2486, 5455, -5451, 3272, -2195, 1365, -897, 693, -123, -130, 167, -182, 122, -52, 21, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 7*x^19 + 21*x^18 - 52*x^17 + 122*x^16 - 182*x^15 + 167*x^14 - 130*x^13 - 123*x^12 + 693*x^11 - 897*x^10 + 1365*x^9 - 2195*x^8 + 3272*x^7 - 5451*x^6 + 5455*x^5 - 2486*x^4 + 652*x^3 - 368*x^2 + 176*x - 32)
 
gp: K = bnfinit(x^20 - 7*x^19 + 21*x^18 - 52*x^17 + 122*x^16 - 182*x^15 + 167*x^14 - 130*x^13 - 123*x^12 + 693*x^11 - 897*x^10 + 1365*x^9 - 2195*x^8 + 3272*x^7 - 5451*x^6 + 5455*x^5 - 2486*x^4 + 652*x^3 - 368*x^2 + 176*x - 32, 1)
 

Normalized defining polynomial

\( x^{20} - 7 x^{19} + 21 x^{18} - 52 x^{17} + 122 x^{16} - 182 x^{15} + 167 x^{14} - 130 x^{13} - 123 x^{12} + 693 x^{11} - 897 x^{10} + 1365 x^{9} - 2195 x^{8} + 3272 x^{7} - 5451 x^{6} + 5455 x^{5} - 2486 x^{4} + 652 x^{3} - 368 x^{2} + 176 x - 32 \)

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:  \(-6965462128295683654948996096=-\,2^{10}\cdot 11^{16}\cdot 23^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $24.67$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{15} - \frac{1}{2} a^{14} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{17} - \frac{1}{4} a^{16} - \frac{1}{4} a^{15} - \frac{1}{2} a^{14} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{4} a^{11} + \frac{1}{4} a^{9} - \frac{1}{4} a^{8} + \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{88} a^{18} + \frac{3}{88} a^{17} + \frac{15}{88} a^{16} - \frac{5}{44} a^{15} + \frac{5}{44} a^{14} + \frac{7}{44} a^{13} + \frac{35}{88} a^{12} - \frac{5}{22} a^{11} + \frac{1}{88} a^{10} + \frac{15}{88} a^{9} + \frac{9}{88} a^{8} + \frac{35}{88} a^{7} + \frac{31}{88} a^{6} + \frac{17}{44} a^{5} + \frac{21}{88} a^{4} + \frac{41}{88} a^{3} - \frac{2}{11} a^{2} - \frac{2}{11} a - \frac{5}{11}$, $\frac{1}{4881703391253072559008261296} a^{19} - \frac{12093174460181294236827149}{4881703391253072559008261296} a^{18} + \frac{43721784104114165221361081}{443791217386642959909841936} a^{17} - \frac{142915746719834756786427115}{2440851695626536279504130648} a^{16} + \frac{198362802762957719100968135}{2440851695626536279504130648} a^{15} + \frac{750781823480456064976254283}{2440851695626536279504130648} a^{14} - \frac{321141102502299262011605933}{4881703391253072559008261296} a^{13} + \frac{346903304101665456957468655}{1220425847813268139752065324} a^{12} - \frac{2072349765925874239547819107}{4881703391253072559008261296} a^{11} - \frac{1793538358338239357232767721}{4881703391253072559008261296} a^{10} - \frac{191378392361508101779698849}{443791217386642959909841936} a^{9} - \frac{1995636206581144719946776377}{4881703391253072559008261296} a^{8} - \frac{1904328350392778164360044045}{4881703391253072559008261296} a^{7} + \frac{75878031254560656829940625}{221895608693321479954920968} a^{6} - \frac{2429463670105750856652159311}{4881703391253072559008261296} a^{5} - \frac{20247876189941957778280695}{4881703391253072559008261296} a^{4} + \frac{557234916141994977554386397}{1220425847813268139752065324} a^{3} + \frac{443953831962871591534393445}{1220425847813268139752065324} a^{2} + \frac{289057907143881323706089353}{610212923906634069876032662} a - \frac{18538117879031922047992504}{305106461953317034938016331}$

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:  \( 1409730.32164 \) (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

\(\Q(\zeta_{11})^+\), 10.6.113395848049.1

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 ${\href{/LocalNumberField/3.10.0.1}{10} }{,}\,{\href{/LocalNumberField/3.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }^{2}$ R ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }^{2}$ R ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }^{2}$

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
$2$2.5.0.1$x^{5} + x^{2} + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
2.5.0.1$x^{5} + x^{2} + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
2.10.10.7$x^{10} - x^{8} - x^{6} - 3 x^{2} - 7$$2$$5$$10$$C_{10}$$[2]^{5}$
$11$11.5.4.4$x^{5} - 11$$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$$0$Trivial$[\ ]$
23.2.0.1$x^{2} - x + 7$$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}$
23.4.3.1$x^{4} + 46$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
23.4.3.1$x^{4} + 46$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$