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SageMath
E = EllipticCurve("x1")
E.isogeny_class()
Elliptic curves in class 81312x
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 81312.bh3 | 81312x1 | \([0, 1, 0, -1734, 25056]\) | \(5088448/441\) | \(50000537664\) | \([2, 2]\) | \(81920\) | \(0.79374\) | \(\Gamma_0(N)\)-optimal |
| 81312.bh4 | 81312x2 | \([0, 1, 0, 1896, 119436]\) | \(830584/7203\) | \(-6533403588096\) | \([2]\) | \(163840\) | \(1.1403\) | |
| 81312.bh2 | 81312x3 | \([0, 1, 0, -5969, -150273]\) | \(3241792/567\) | \(4114329956352\) | \([2]\) | \(163840\) | \(1.1403\) | |
| 81312.bh1 | 81312x4 | \([0, 1, 0, -27144, 1712280]\) | \(2438569736/21\) | \(19047823872\) | \([2]\) | \(163840\) | \(1.1403\) |
Rank
sage: E.rank()
The elliptic curves in class 81312x have rank \(0\).
Complex multiplication
The elliptic curves in class 81312x do not have complex multiplication.Modular form 81312.2.a.x
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 2 & 2 \\ 2 & 1 & 4 & 4 \\ 2 & 4 & 1 & 4 \\ 2 & 4 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
