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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 19320e
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 19320.j3 | 19320e1 | \([0, -1, 0, -186655, 31021900]\) | \(44949507773962418176/132895751953125\) | \(2126332031250000\) | \([4]\) | \(138240\) | \(1.8110\) | \(\Gamma_0(N)\)-optimal |
| 19320.j2 | 19320e2 | \([0, -1, 0, -264780, 2646900]\) | \(8019382352783901136/4629798816890625\) | \(1185228497124000000\) | \([2, 2]\) | \(276480\) | \(2.1576\) | |
| 19320.j1 | 19320e3 | \([0, -1, 0, -2837280, -1832060100]\) | \(2466780454987534385284/10072750481768625\) | \(10314496493331072000\) | \([2]\) | \(552960\) | \(2.5041\) | |
| 19320.j4 | 19320e4 | \([0, -1, 0, 1057720, 20103900]\) | \(127801365439147434716/74135664409456125\) | \(-75914920355283072000\) | \([2]\) | \(552960\) | \(2.5041\) |
Rank
sage: E.rank()
The elliptic curves in class 19320e have rank \(1\).
Complex multiplication
The elliptic curves in class 19320e do not have complex multiplication.Modular form 19320.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
