Show commands:
SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 576f
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality | CM discriminant |
---|---|---|---|---|---|---|---|---|---|
576.i1 | 576f1 | \([0, 0, 0, -3, 0]\) | \(1728\) | \(1728\) | \([2]\) | \(32\) | \(-0.68931\) | \(\Gamma_0(N)\)-optimal | \(-4\) |
576.i2 | 576f2 | \([0, 0, 0, 12, 0]\) | \(1728\) | \(-110592\) | \([2]\) | \(64\) | \(-0.34273\) | \(-4\) |
Rank
sage: E.rank()
The elliptic curves in class 576f have rank \(0\).
Complex multiplication
Each elliptic curve in class 576f has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-1}) \).Modular form 576.2.a.f
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.