↑

Properties

Label 23.1.18234914707...9779.1
Degree $23$
Signature $[1, 11]$
Discriminant $-\,1979^{11}$
Root discriminant $37.72$
Ramified prime $1979$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $D_{23}$ (as 23T2)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-2176, 1488, 2872, -2608, -6818, 10047, -125, -8091, 3246, 5059, -5271, -393, 3310, -1586, -712, 1182, -364, -359, 367, -51, -78, 47, -11, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^23 - 11*x^22 + 47*x^21 - 78*x^20 - 51*x^19 + 367*x^18 - 359*x^17 - 364*x^16 + 1182*x^15 - 712*x^14 - 1586*x^13 + 3310*x^12 - 393*x^11 - 5271*x^10 + 5059*x^9 + 3246*x^8 - 8091*x^7 - 125*x^6 + 10047*x^5 - 6818*x^4 - 2608*x^3 + 2872*x^2 + 1488*x - 2176)
 
gp: K = bnfinit(x^23 - 11*x^22 + 47*x^21 - 78*x^20 - 51*x^19 + 367*x^18 - 359*x^17 - 364*x^16 + 1182*x^15 - 712*x^14 - 1586*x^13 + 3310*x^12 - 393*x^11 - 5271*x^10 + 5059*x^9 + 3246*x^8 - 8091*x^7 - 125*x^6 + 10047*x^5 - 6818*x^4 - 2608*x^3 + 2872*x^2 + 1488*x - 2176, 1)
 

Normalized defining polynomial

\( x^{23} - 11 x^{22} + 47 x^{21} - 78 x^{20} - 51 x^{19} + 367 x^{18} - 359 x^{17} - 364 x^{16} + 1182 x^{15} - 712 x^{14} - 1586 x^{13} + 3310 x^{12} - 393 x^{11} - 5271 x^{10} + 5059 x^{9} + 3246 x^{8} - 8091 x^{7} - 125 x^{6} + 10047 x^{5} - 6818 x^{4} - 2608 x^{3} + 2872 x^{2} + 1488 x - 2176 \)

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

Invariants

Degree:  $23$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[1, 11]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-1823491470737247969166873376199879779=-\,1979^{11}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $37.72$
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:  $1979$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{3}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{4}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{5}$, $\frac{1}{8} a^{12} - \frac{1}{8} a^{9} - \frac{1}{8} a^{6} + \frac{1}{8} a^{3}$, $\frac{1}{16} a^{13} - \frac{1}{8} a^{11} - \frac{1}{16} a^{10} - \frac{1}{8} a^{9} + \frac{3}{16} a^{7} + \frac{1}{8} a^{5} - \frac{3}{16} a^{4} - \frac{3}{8} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{16} a^{14} - \frac{1}{16} a^{11} - \frac{1}{8} a^{10} - \frac{1}{8} a^{9} - \frac{1}{16} a^{8} - \frac{3}{16} a^{5} + \frac{1}{8} a^{4} - \frac{3}{8} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{16} a^{15} - \frac{1}{16} a^{12} - \frac{1}{8} a^{11} - \frac{1}{8} a^{10} - \frac{1}{16} a^{9} - \frac{3}{16} a^{6} + \frac{1}{8} a^{5} + \frac{1}{8} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{32} a^{16} + \frac{1}{16} a^{10} - \frac{1}{4} a^{7} - \frac{3}{32} a^{4} + \frac{1}{4} a$, $\frac{1}{32} a^{17} + \frac{1}{16} a^{11} - \frac{3}{32} a^{5}$, $\frac{1}{32} a^{18} - \frac{1}{16} a^{12} - \frac{1}{8} a^{9} + \frac{1}{32} a^{6} + \frac{1}{8} a^{3}$, $\frac{1}{32} a^{19} - \frac{1}{8} a^{11} + \frac{1}{16} a^{10} - \frac{1}{8} a^{9} + \frac{7}{32} a^{7} + \frac{1}{8} a^{5} + \frac{3}{16} a^{4} - \frac{3}{8} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{64} a^{20} - \frac{1}{64} a^{17} - \frac{1}{32} a^{14} - \frac{3}{32} a^{11} - \frac{1}{8} a^{10} - \frac{1}{8} a^{9} - \frac{7}{64} a^{8} - \frac{9}{64} a^{5} + \frac{1}{8} a^{4} - \frac{3}{8} a^{3} + \frac{3}{8} a^{2} - \frac{1}{2} a$, $\frac{1}{2542991104} a^{21} + \frac{15711945}{2542991104} a^{20} - \frac{85999}{5941568} a^{19} + \frac{38623339}{2542991104} a^{18} - \frac{20830419}{2542991104} a^{17} - \frac{2662609}{1271495552} a^{16} - \frac{2711505}{1271495552} a^{15} - \frac{7851203}{1271495552} a^{14} + \frac{9104205}{635747776} a^{13} - \frac{10476871}{1271495552} a^{12} + \frac{127819749}{1271495552} a^{11} + \frac{31162009}{317873888} a^{10} + \frac{225510353}{2542991104} a^{9} - \frac{81803051}{2542991104} a^{8} - \frac{37056}{584327} a^{7} + \frac{53996883}{2542991104} a^{6} + \frac{140776233}{2542991104} a^{5} + \frac{230558005}{1271495552} a^{4} - \frac{15188691}{39734236} a^{3} + \frac{67112569}{317873888} a^{2} + \frac{5690731}{158936944} a + \frac{399831}{1168654}$, $\frac{1}{7223994354800670208} a^{22} - \frac{32236769}{3611997177400335104} a^{21} - \frac{12480890418743055}{7223994354800670208} a^{20} - \frac{3651022161665313}{7223994354800670208} a^{19} + \frac{595055344691565}{106235211100009856} a^{18} - \frac{68544658981136753}{7223994354800670208} a^{17} + \frac{10882727393095497}{1805998588700167552} a^{16} - \frac{1727189414823665}{56437455896880236} a^{15} - \frac{2276908766456481}{97621545335144192} a^{14} + \frac{91510751694803347}{3611997177400335104} a^{13} + \frac{81620667200243501}{1805998588700167552} a^{12} - \frac{28554517181878747}{3611997177400335104} a^{11} + \frac{801576953185247129}{7223994354800670208} a^{10} + \frac{142761448428837765}{3611997177400335104} a^{9} - \frac{392577123916385103}{7223994354800670208} a^{8} + \frac{1800060526878617803}{7223994354800670208} a^{7} - \frac{75699252916715915}{902999294350083776} a^{6} + \frac{1349702774902286231}{7223994354800670208} a^{5} + \frac{849920503594320641}{3611997177400335104} a^{4} - \frac{72441255508022767}{902999294350083776} a^{3} - \frac{100456501116197669}{902999294350083776} a^{2} + \frac{88677540986601359}{451499647175041888} a + \frac{628475915758957}{3319850346875308}$

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:  \( 15406380217.0 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_{23}$ (as 23T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 46
The 13 conjugacy class representatives for $D_{23}$
Character table for $D_{23}$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

Sibling fields

Galois closure: 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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{11}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }$ $23$ $23$ $23$ $23$ $23$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{11}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ $23$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{11}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ $23$ $23$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{11}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{11}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{11}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ $23$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{11}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ $23$

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
1979Data not computed