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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 137904.d
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 137904.d1 | 137904ba2 | \([0, -1, 0, -484072, 155404144]\) | \(-3754462153/943296\) | \(-3151771755300519936\) | \([]\) | \(2695680\) | \(2.2673\) | |
| 137904.d2 | 137904ba1 | \([0, -1, 0, 43208, -1514384]\) | \(2669927/1836\) | \(-6134503848984576\) | \([]\) | \(898560\) | \(1.7180\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 137904.d have rank \(2\).
Complex multiplication
The elliptic curves in class 137904.d do not have complex multiplication.Modular form 137904.2.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.
