↑

Properties

Label 20.4.92832438244...4869.1
Degree $20$
Signature $[4, 8]$
Discriminant $11^{16}\cdot 23^{2}\cdot 1563101^{3}$
Root discriminant $79.14$
Ramified primes $11, 23, 1563101$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T335

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![22936847, -84055216, 122959250, -81796339, 10928294, 17754166, -10762044, 2687586, -517848, -280392, 390034, -105035, -5535, 6576, -6012, 3341, -601, 106, -27, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 - 27*x^18 + 106*x^17 - 601*x^16 + 3341*x^15 - 6012*x^14 + 6576*x^13 - 5535*x^12 - 105035*x^11 + 390034*x^10 - 280392*x^9 - 517848*x^8 + 2687586*x^7 - 10762044*x^6 + 17754166*x^5 + 10928294*x^4 - 81796339*x^3 + 122959250*x^2 - 84055216*x + 22936847)
 
gp: K = bnfinit(x^20 - 3*x^19 - 27*x^18 + 106*x^17 - 601*x^16 + 3341*x^15 - 6012*x^14 + 6576*x^13 - 5535*x^12 - 105035*x^11 + 390034*x^10 - 280392*x^9 - 517848*x^8 + 2687586*x^7 - 10762044*x^6 + 17754166*x^5 + 10928294*x^4 - 81796339*x^3 + 122959250*x^2 - 84055216*x + 22936847, 1)
 

Normalized defining polynomial

\( x^{20} - 3 x^{19} - 27 x^{18} + 106 x^{17} - 601 x^{16} + 3341 x^{15} - 6012 x^{14} + 6576 x^{13} - 5535 x^{12} - 105035 x^{11} + 390034 x^{10} - 280392 x^{9} - 517848 x^{8} + 2687586 x^{7} - 10762044 x^{6} + 17754166 x^{5} + 10928294 x^{4} - 81796339 x^{3} + 122959250 x^{2} - 84055216 x + 22936847 \)

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:  \(92832438244258564416940225807466334869=11^{16}\cdot 23^{2}\cdot 1563101^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $79.14$
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:  $11, 23, 1563101$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{5} a^{15} + \frac{1}{5} a^{13} - \frac{1}{5} a^{12} - \frac{1}{5} a^{9} - \frac{1}{5} a^{8} + \frac{2}{5} a^{7} - \frac{2}{5} a^{6} - \frac{2}{5} a^{5} - \frac{2}{5} a^{4} + \frac{2}{5} a^{2} + \frac{1}{5} a - \frac{1}{5}$, $\frac{1}{5} a^{16} + \frac{1}{5} a^{14} - \frac{1}{5} a^{13} - \frac{1}{5} a^{10} - \frac{1}{5} a^{9} + \frac{2}{5} a^{8} - \frac{2}{5} a^{7} - \frac{2}{5} a^{6} - \frac{2}{5} a^{5} + \frac{2}{5} a^{3} + \frac{1}{5} a^{2} - \frac{1}{5} a$, $\frac{1}{5} a^{17} - \frac{1}{5} a^{14} - \frac{1}{5} a^{13} + \frac{1}{5} a^{12} - \frac{1}{5} a^{11} - \frac{1}{5} a^{10} - \frac{2}{5} a^{9} - \frac{1}{5} a^{8} + \frac{1}{5} a^{7} + \frac{2}{5} a^{5} - \frac{1}{5} a^{4} + \frac{1}{5} a^{3} + \frac{2}{5} a^{2} - \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{18} - \frac{1}{5} a^{14} + \frac{2}{5} a^{13} - \frac{2}{5} a^{12} - \frac{1}{5} a^{11} - \frac{2}{5} a^{10} - \frac{2}{5} a^{9} + \frac{2}{5} a^{7} + \frac{2}{5} a^{5} - \frac{1}{5} a^{4} + \frac{2}{5} a^{3} + \frac{1}{5} a^{2} + \frac{2}{5} a - \frac{1}{5}$, $\frac{1}{55762856215851367773357213373762153568753065252099239822067605860165} a^{19} - \frac{2996480986352889961104743308196473080983631771391252981613062162206}{55762856215851367773357213373762153568753065252099239822067605860165} a^{18} + \frac{1979388732308646523178892946667476923057751461264820532934760798771}{55762856215851367773357213373762153568753065252099239822067605860165} a^{17} - \frac{5336063857575947774540869373495394589519831167999374780355358134064}{55762856215851367773357213373762153568753065252099239822067605860165} a^{16} - \frac{1036677115021966805441149240921051853931268456620114345326905721524}{11152571243170273554671442674752430713750613050419847964413521172033} a^{15} - \frac{16614658780241799568863629905702386666745162731246811784170367220737}{55762856215851367773357213373762153568753065252099239822067605860165} a^{14} + \frac{823131326082216447146765264907412686807835638486796502706238656396}{11152571243170273554671442674752430713750613050419847964413521172033} a^{13} - \frac{24174754262129797078727011675503493032667582619402025895709432773259}{55762856215851367773357213373762153568753065252099239822067605860165} a^{12} - \frac{12074111802520518539915491940690444617318350195633235733003384203717}{55762856215851367773357213373762153568753065252099239822067605860165} a^{11} + \frac{13095727539871942929963295904357942548623389480147532110867082993323}{55762856215851367773357213373762153568753065252099239822067605860165} a^{10} - \frac{2348484884410988116246707134576280194260008675042755923295809083847}{55762856215851367773357213373762153568753065252099239822067605860165} a^{9} + \frac{3880588908208630959857101094440682326860058304124808291691050541767}{55762856215851367773357213373762153568753065252099239822067605860165} a^{8} - \frac{11976484679592149468535209935035566633900989099758437853838620308921}{55762856215851367773357213373762153568753065252099239822067605860165} a^{7} + \frac{15731167721825750018363144007671986795444385259469047220649730109938}{55762856215851367773357213373762153568753065252099239822067605860165} a^{6} + \frac{1451215778001803103961944776189879968953021939281849330452985406817}{11152571243170273554671442674752430713750613050419847964413521172033} a^{5} + \frac{2440717968500845708348124248237715298507418090318983440020207635626}{11152571243170273554671442674752430713750613050419847964413521172033} a^{4} - \frac{16767837578772320518170214430621971924508153589729268097318173824678}{55762856215851367773357213373762153568753065252099239822067605860165} a^{3} - \frac{16455393770055531584187213573436453078602820822627214033552725712189}{55762856215851367773357213373762153568753065252099239822067605860165} a^{2} + \frac{15940140793792076726100729155993312333763118841489667981374994705236}{55762856215851367773357213373762153568753065252099239822067605860165} a - \frac{11814651577117787845089373562003013771294124519199287446890775207659}{55762856215851367773357213373762153568753065252099239822067605860165}$

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:  \( 110677029457 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T335:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 5120
The 224 conjugacy class representatives for t20n335 are not computed
Character table for t20n335 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.10.335064581249981.1

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 $20$ $20$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ R $20$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{8}$ $20$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ $20$

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
$11$11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$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$$0$Trivial$[\ ]$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
23.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
23.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
1563101Data not computed