Show commands:
SageMath
E = EllipticCurve("g1")
E.isogeny_class()
Elliptic curves in class 3872.g
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality | CM discriminant |
---|---|---|---|---|---|---|---|---|---|
3872.g1 | 3872h2 | \([0, 0, 0, -44, 0]\) | \(1728\) | \(5451776\) | \([2]\) | \(480\) | \(-0.017912\) | \(-4\) | |
3872.g2 | 3872h1 | \([0, 0, 0, 11, 0]\) | \(1728\) | \(-85184\) | \([2]\) | \(240\) | \(-0.36449\) | \(\Gamma_0(N)\)-optimal | \(-4\) |
Rank
sage: E.rank()
The elliptic curves in class 3872.g have rank \(0\).
Complex multiplication
Each elliptic curve in class 3872.g has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-1}) \).Modular form 3872.2.a.g
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.