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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 768.e
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 768.e1 | 768h4 | \([0, 1, 0, -2589, 49851]\) | \(58591911104/243\) | \(7962624\) | \([2]\) | \(320\) | \(0.53358\) | |
| 768.e2 | 768h3 | \([0, 1, 0, -159, 765]\) | \(-873722816/59049\) | \(-30233088\) | \([2]\) | \(160\) | \(0.18700\) | |
| 768.e3 | 768h2 | \([0, 1, 0, -29, -69]\) | \(85184/3\) | \(98304\) | \([2]\) | \(64\) | \(-0.27114\) | |
| 768.e4 | 768h1 | \([0, 1, 0, 1, -3]\) | \(64/9\) | \(-4608\) | \([2]\) | \(32\) | \(-0.61772\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 768.e have rank \(1\).
Complex multiplication
The elliptic curves in class 768.e do not have complex multiplication.Modular form 768.2.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 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.
