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SageMath
E = EllipticCurve("gp1")
E.isogeny_class()
Elliptic curves in class 78400.gp
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 78400.gp1 | 78400gx4 | \([0, 0, 0, -524300, -146118000]\) | \(132304644/5\) | \(602362880000000\) | \([2]\) | \(442368\) | \(1.9220\) | |
| 78400.gp2 | 78400gx2 | \([0, 0, 0, -34300, -2058000]\) | \(148176/25\) | \(752953600000000\) | \([2, 2]\) | \(221184\) | \(1.5755\) | |
| 78400.gp3 | 78400gx1 | \([0, 0, 0, -9800, 343000]\) | \(55296/5\) | \(9411920000000\) | \([2]\) | \(110592\) | \(1.2289\) | \(\Gamma_0(N)\)-optimal |
| 78400.gp4 | 78400gx3 | \([0, 0, 0, 63700, -11662000]\) | \(237276/625\) | \(-75295360000000000\) | \([2]\) | \(442368\) | \(1.9220\) |
Rank
sage: E.rank()
The elliptic curves in class 78400.gp have rank \(1\).
Complex multiplication
The elliptic curves in class 78400.gp do not have complex multiplication.Modular form 78400.2.a.gp
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
