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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 1335.a
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 1335.a1 | 1335b3 | \([1, 0, 0, -2375, -44748]\) | \(1481582988342001/2919645\) | \(2919645\) | \([2]\) | \(896\) | \(0.49306\) | |
| 1335.a2 | 1335b2 | \([1, 0, 0, -150, -693]\) | \(373403541601/16040025\) | \(16040025\) | \([2, 2]\) | \(448\) | \(0.14649\) | |
| 1335.a3 | 1335b1 | \([1, 0, 0, -25, 32]\) | \(1732323601/500625\) | \(500625\) | \([4]\) | \(224\) | \(-0.20008\) | \(\Gamma_0(N)\)-optimal |
| 1335.a4 | 1335b4 | \([1, 0, 0, 75, -2538]\) | \(46617130799/2823400845\) | \(-2823400845\) | \([2]\) | \(896\) | \(0.49306\) |
Rank
sage: E.rank()
The elliptic curves in class 1335.a have rank \(0\).
Complex multiplication
The elliptic curves in class 1335.a do not have complex multiplication.Modular form 1335.2.a.a
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 LMFDB 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 LMFDB labels.
