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SageMath
E = EllipticCurve("eu1")
E.isogeny_class()
Elliptic curves in class 77616eu
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 77616.dk4 | 77616eu1 | \([0, 0, 0, 543165, -91622846]\) | \(50447927375/39517632\) | \(-13882466335725453312\) | \([2]\) | \(884736\) | \(2.3611\) | \(\Gamma_0(N)\)-optimal |
| 77616.dk3 | 77616eu2 | \([0, 0, 0, -2561475, -791408702]\) | \(5290763640625/2291573592\) | \(805025292172806684672\) | \([2]\) | \(1769472\) | \(2.7077\) | |
| 77616.dk2 | 77616eu3 | \([0, 0, 0, -5807235, 6841235842]\) | \(-61653281712625/21875235228\) | \(-7684727076733380968448\) | \([2]\) | \(2654208\) | \(2.9104\) | |
| 77616.dk1 | 77616eu4 | \([0, 0, 0, -99722595, 383272781794]\) | \(312196988566716625/25367712678\) | \(8911627530381648027648\) | \([2]\) | \(5308416\) | \(3.2570\) |
Rank
sage: E.rank()
The elliptic curves in class 77616eu have rank \(0\).
Complex multiplication
The elliptic curves in class 77616eu do not have complex multiplication.Modular form 77616.2.a.eu
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
