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SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 675c
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality | CM discriminant |
|---|---|---|---|---|---|---|---|---|---|
| 675.d2 | 675c1 | \([0, 0, 1, 0, 6]\) | \(0\) | \(-16875\) | \([3]\) | \(42\) | \(-0.50998\) | \(\Gamma_0(N)\)-optimal | \(-3\) |
| 675.d1 | 675c2 | \([0, 0, 1, 0, -169]\) | \(0\) | \(-12301875\) | \([]\) | \(126\) | \(0.039321\) | \(-3\) |
Rank
sage: E.rank()
The elliptic curves in class 675c have rank \(0\).
Complex multiplication
Each elliptic curve in class 675c has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-3}) \).Modular form 675.2.a.c
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
