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SageMath
E = EllipticCurve("dt1")
E.isogeny_class()
Elliptic curves in class 86640.dt
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 86640.dt1 | 86640ba6 | \([0, 1, 0, -1155320, -478356300]\) | \(1770025017602/75\) | \(7226247321600\) | \([2]\) | \(884736\) | \(1.9522\) | |
| 86640.dt2 | 86640ba4 | \([0, 1, 0, -72320, -7467900]\) | \(868327204/5625\) | \(270984274560000\) | \([2, 2]\) | \(442368\) | \(1.6056\) | |
| 86640.dt3 | 86640ba5 | \([0, 1, 0, -29000, -16287852]\) | \(-27995042/1171875\) | \(-112910114400000000\) | \([2]\) | \(884736\) | \(1.9522\) | |
| 86640.dt4 | 86640ba2 | \([0, 1, 0, -7340, 43788]\) | \(3631696/2025\) | \(24388584710400\) | \([2, 2]\) | \(221184\) | \(1.2591\) | |
| 86640.dt5 | 86640ba1 | \([0, 1, 0, -5535, 156420]\) | \(24918016/45\) | \(33873034320\) | \([2]\) | \(110592\) | \(0.91250\) | \(\Gamma_0(N)\)-optimal |
| 86640.dt6 | 86640ba3 | \([0, 1, 0, 28760, 375908]\) | \(54607676/32805\) | \(-1580380289233920\) | \([2]\) | \(442368\) | \(1.6056\) |
Rank
sage: E.rank()
The elliptic curves in class 86640.dt have rank \(0\).
Complex multiplication
The elliptic curves in class 86640.dt do not have complex multiplication.Modular form 86640.2.a.dt
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 LMFDB numbering.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
