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SageMath
E = EllipticCurve("l1")
E.isogeny_class()
Elliptic curves in class 12138l
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 12138.i3 | 12138l1 | \([1, 0, 1, -542, 4880]\) | \(-3574558889/64512\) | \(-316947456\) | \([2]\) | \(6400\) | \(0.42751\) | \(\Gamma_0(N)\)-optimal |
| 12138.i2 | 12138l2 | \([1, 0, 1, -8702, 311696]\) | \(14830727012009/4704\) | \(23110752\) | \([2]\) | \(12800\) | \(0.77408\) | |
| 12138.i4 | 12138l3 | \([1, 0, 1, 3198, -200480]\) | \(736558976791/3969746172\) | \(-19503362943036\) | \([2]\) | \(32000\) | \(1.2322\) | |
| 12138.i1 | 12138l4 | \([1, 0, 1, -38112, -2579936]\) | \(1246079601667529/137282971014\) | \(674471236591782\) | \([2]\) | \(64000\) | \(1.5788\) |
Rank
sage: E.rank()
The elliptic curves in class 12138l have rank \(1\).
Complex multiplication
The elliptic curves in class 12138l do not have complex multiplication.Modular form 12138.2.a.l
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 5 & 10 \\ 2 & 1 & 10 & 5 \\ 5 & 10 & 1 & 2 \\ 10 & 5 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
