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SageMath
E = EllipticCurve("ef1")
E.isogeny_class()
Elliptic curves in class 103488ef
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 103488.fo3 | 103488ef1 | \([0, 1, 0, -6484, -203134]\) | \(4004529472/99\) | \(745424064\) | \([2]\) | \(98304\) | \(0.81160\) | \(\Gamma_0(N)\)-optimal |
| 103488.fo2 | 103488ef2 | \([0, 1, 0, -6729, -187209]\) | \(69934528/9801\) | \(4723006869504\) | \([2, 2]\) | \(196608\) | \(1.1582\) | |
| 103488.fo4 | 103488ef3 | \([0, 1, 0, 10911, -988065]\) | \(37259704/131769\) | \(-507985627742208\) | \([2]\) | \(393216\) | \(1.5047\) | |
| 103488.fo1 | 103488ef4 | \([0, 1, 0, -28289, 1636767]\) | \(649461896/72171\) | \(278228041039872\) | \([2]\) | \(393216\) | \(1.5047\) |
Rank
sage: E.rank()
The elliptic curves in class 103488ef have rank \(2\).
Complex multiplication
The elliptic curves in class 103488ef do not have complex multiplication.Modular form 103488.2.a.ef
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.
