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SageMath
E = EllipticCurve("dp1")
E.isogeny_class()
Elliptic curves in class 25200.dp
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 25200.dp1 | 25200fr2 | \([0, 0, 0, -534000, 150950000]\) | \(-2887553024/16807\) | \(-98018424000000000\) | \([]\) | \(240000\) | \(2.1014\) | |
| 25200.dp2 | 25200fr1 | \([0, 0, 0, 6000, -250000]\) | \(4096/7\) | \(-40824000000000\) | \([]\) | \(48000\) | \(1.2966\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 25200.dp have rank \(1\).
Complex multiplication
The elliptic curves in class 25200.dp do not have complex multiplication.Modular form 25200.2.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 LMFDB numbering.
\(\left(\begin{array}{rr} 1 & 5 \\ 5 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
