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SageMath
E = EllipticCurve("t1")
E.isogeny_class()
Elliptic curves in class 6400t
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality | CM discriminant |
|---|---|---|---|---|---|---|---|---|---|
| 6400.k1 | 6400t1 | \([0, 0, 0, -10, 0]\) | \(1728\) | \(64000\) | \([2]\) | \(384\) | \(-0.38831\) | \(\Gamma_0(N)\)-optimal | \(-4\) |
| 6400.k2 | 6400t2 | \([0, 0, 0, 40, 0]\) | \(1728\) | \(-4096000\) | \([2]\) | \(768\) | \(-0.041739\) | \(-4\) |
Rank
sage: E.rank()
The elliptic curves in class 6400t have rank \(1\).
Complex multiplication
Each elliptic curve in class 6400t has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-1}) \).Modular form 6400.2.a.t
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.
