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SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 3192j
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 3192.h3 | 3192j1 | \([0, -1, 0, -932, 11268]\) | \(350104249168/2793\) | \(715008\) | \([4]\) | \(1280\) | \(0.29545\) | \(\Gamma_0(N)\)-optimal |
| 3192.h2 | 3192j2 | \([0, -1, 0, -952, 10780]\) | \(93280467172/7800849\) | \(7988069376\) | \([2, 2]\) | \(2560\) | \(0.64202\) | |
| 3192.h1 | 3192j3 | \([0, -1, 0, -3232, -57620]\) | \(1823652903746/328593657\) | \(672959809536\) | \([2]\) | \(5120\) | \(0.98860\) | |
| 3192.h4 | 3192j4 | \([0, -1, 0, 1008, 47628]\) | \(55251546334/517244049\) | \(-1059315812352\) | \([2]\) | \(5120\) | \(0.98860\) |
Rank
sage: E.rank()
The elliptic curves in class 3192j have rank \(0\).
Complex multiplication
The elliptic curves in class 3192j do not have complex multiplication.Modular form 3192.2.a.j
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.
