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SageMath
E = EllipticCurve("bj1")
E.isogeny_class()
Elliptic curves in class 34650bj
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 34650.bw4 | 34650bj1 | \([1, -1, 0, -214317, -51069659]\) | \(-95575628340361/43812679680\) | \(-499053804480000000\) | \([2]\) | \(589824\) | \(2.1024\) | \(\Gamma_0(N)\)-optimal |
| 34650.bw3 | 34650bj2 | \([1, -1, 0, -3742317, -2785269659]\) | \(508859562767519881/62240270400\) | \(708955580025000000\) | \([2, 2]\) | \(1179648\) | \(2.4490\) | |
| 34650.bw2 | 34650bj3 | \([1, -1, 0, -4057317, -2288514659]\) | \(648474704552553481/176469171805080\) | \(2010094160092239375000\) | \([2]\) | \(2359296\) | \(2.7956\) | |
| 34650.bw1 | 34650bj4 | \([1, -1, 0, -59875317, -178313160659]\) | \(2084105208962185000201/31185000\) | \(355216640625000\) | \([2]\) | \(2359296\) | \(2.7956\) |
Rank
sage: E.rank()
The elliptic curves in class 34650bj have rank \(0\).
Complex multiplication
The elliptic curves in class 34650bj do not have complex multiplication.Modular form 34650.2.a.bj
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
