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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 720e
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 720.f4 | 720e1 | \([0, 0, 0, 33, -34]\) | \(21296/15\) | \(-2799360\) | \([2]\) | \(128\) | \(-0.074438\) | \(\Gamma_0(N)\)-optimal |
| 720.f3 | 720e2 | \([0, 0, 0, -147, -286]\) | \(470596/225\) | \(167961600\) | \([2, 2]\) | \(256\) | \(0.27214\) | |
| 720.f1 | 720e3 | \([0, 0, 0, -1947, -33046]\) | \(546718898/405\) | \(604661760\) | \([2]\) | \(512\) | \(0.61871\) | |
| 720.f2 | 720e4 | \([0, 0, 0, -1227, 16346]\) | \(136835858/1875\) | \(2799360000\) | \([4]\) | \(512\) | \(0.61871\) |
Rank
sage: E.rank()
The elliptic curves in class 720e have rank \(1\).
Complex multiplication
The elliptic curves in class 720e do not have complex multiplication.Modular form 720.2.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 Cremona numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
