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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 34969e
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 34969.l3 | 34969e1 | \([1, -1, 0, -24041, -607040]\) | \(35937/17\) | \(726939989878553\) | \([2]\) | \(103680\) | \(1.5458\) | \(\Gamma_0(N)\)-optimal |
| 34969.l2 | 34969e2 | \([1, -1, 0, -198886, 33767487]\) | \(20346417/289\) | \(12357979827935401\) | \([2, 2]\) | \(207360\) | \(1.8923\) | |
| 34969.l4 | 34969e3 | \([1, -1, 0, -24041, 90941802]\) | \(-35937/83521\) | \(-3571456170273330889\) | \([2]\) | \(414720\) | \(2.2389\) | |
| 34969.l1 | 34969e4 | \([1, -1, 0, -3171251, 2174464760]\) | \(82483294977/17\) | \(726939989878553\) | \([2]\) | \(414720\) | \(2.2389\) |
Rank
sage: E.rank()
The elliptic curves in class 34969e have rank \(0\).
Complex multiplication
The elliptic curves in class 34969e do not have complex multiplication.Modular form 34969.2.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.
