Show commands:
SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 384813e
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 384813.e6 | 384813e1 | \([1, -1, 1, 47119, 11530640]\) | \(3288008303/18259263\) | \(-64249667761708143\) | \([2]\) | \(2359296\) | \(1.9072\) | \(\Gamma_0(N)\)-optimal |
| 384813.e5 | 384813e2 | \([1, -1, 1, -568886, 149022956]\) | \(5786435182177/627352209\) | \(2207491671368318049\) | \([2, 2]\) | \(4718592\) | \(2.2537\) | |
| 384813.e2 | 384813e3 | \([1, -1, 1, -8850731, 10136928026]\) | \(21790813729717297/304746849\) | \(1072326073603159089\) | \([2, 2]\) | \(9437184\) | \(2.6003\) | |
| 384813.e4 | 384813e4 | \([1, -1, 1, -2143121, -1046765950]\) | \(309368403125137/44372288367\) | \(156134712852674128287\) | \([2]\) | \(9437184\) | \(2.6003\) | |
| 384813.e1 | 384813e5 | \([1, -1, 1, -141611216, 648661756682]\) | \(89254274298475942657/17457\) | \(61426709835777\) | \([2]\) | \(18874368\) | \(2.9469\) | |
| 384813.e3 | 384813e6 | \([1, -1, 1, -8599766, 10738641710]\) | \(-19989223566735457/2584262514273\) | \(-9093357598884292200753\) | \([2]\) | \(18874368\) | \(2.9469\) |
Rank
sage: E.rank()
The elliptic curves in class 384813e have rank \(2\).
Complex multiplication
The elliptic curves in class 384813e do not have complex multiplication.Modular form 384813.2.a.e
sage: E.q_eigenform(10)
Isogeny matrix
sage: E.isogeny_class().matrix()
The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the Cremona numbering.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
