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SageMath
E = EllipticCurve("dp1")
E.isogeny_class()
Elliptic curves in class 319725dp
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 319725.dp2 | 319725dp1 | \([1, -1, 0, -104967, 13795816]\) | \(-95443993/5887\) | \(-7889143036359375\) | \([2]\) | \(1769472\) | \(1.8048\) | \(\Gamma_0(N)\)-optimal |
| 319725.dp1 | 319725dp2 | \([1, -1, 0, -1703592, 856271191]\) | \(408023180713/1421\) | \(1904275905328125\) | \([2]\) | \(3538944\) | \(2.1513\) |
Rank
sage: E.rank()
The elliptic curves in class 319725dp have rank \(0\).
Complex multiplication
The elliptic curves in class 319725dp do not have complex multiplication.Modular form 319725.2.a.dp
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
