Show commands:
SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 1305.d
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 1305.d1 | 1305f1 | \([0, 0, 1, -102, -405]\) | \(-160989184/3915\) | \(-2854035\) | \([]\) | \(160\) | \(0.024133\) | \(\Gamma_0(N)\)-optimal |
| 1305.d2 | 1305f2 | \([0, 0, 1, 438, -1728]\) | \(12747309056/9145875\) | \(-6667342875\) | \([3]\) | \(480\) | \(0.57344\) |
Rank
sage: E.rank()
The elliptic curves in class 1305.d have rank \(0\).
Complex multiplication
The elliptic curves in class 1305.d do not have complex multiplication.Modular form 1305.2.a.d
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
