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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 768.a
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 768.a1 | 768b4 | \([0, -1, 0, -2589, -49851]\) | \(58591911104/243\) | \(7962624\) | \([2]\) | \(320\) | \(0.53358\) | |
| 768.a2 | 768b3 | \([0, -1, 0, -159, -765]\) | \(-873722816/59049\) | \(-30233088\) | \([2]\) | \(160\) | \(0.18700\) | |
| 768.a3 | 768b2 | \([0, -1, 0, -29, 69]\) | \(85184/3\) | \(98304\) | \([2]\) | \(64\) | \(-0.27114\) | |
| 768.a4 | 768b1 | \([0, -1, 0, 1, 3]\) | \(64/9\) | \(-4608\) | \([2]\) | \(32\) | \(-0.61772\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 768.a have rank \(1\).
Complex multiplication
The elliptic curves in class 768.a do not have complex multiplication.Modular form 768.2.a.a
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 5 & 10 \\ 2 & 1 & 10 & 5 \\ 5 & 10 & 1 & 2 \\ 10 & 5 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
