Show commands:
SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 576.e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality | CM discriminant |
---|---|---|---|---|---|---|---|---|---|
576.e1 | 576a4 | \([0, 0, 0, -540, -4752]\) | \(54000\) | \(322486272\) | \([2]\) | \(192\) | \(0.42704\) | \(-12\) | |
576.e2 | 576a2 | \([0, 0, 0, -60, 176]\) | \(54000\) | \(442368\) | \([2]\) | \(64\) | \(-0.12227\) | \(-12\) | |
576.e3 | 576a3 | \([0, 0, 0, 0, -216]\) | \(0\) | \(-20155392\) | \([2]\) | \(96\) | \(0.080464\) | \(-3\) | |
576.e4 | 576a1 | \([0, 0, 0, 0, 8]\) | \(0\) | \(-27648\) | \([2]\) | \(32\) | \(-0.46884\) | \(\Gamma_0(N)\)-optimal | \(-3\) |
Rank
sage: E.rank()
The elliptic curves in class 576.e have rank \(1\).
Complex multiplication
Each elliptic curve in class 576.e has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-3}) \).Modular form 576.2.a.e
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.
\(\left(\begin{array}{rrrr} 1 & 3 & 2 & 6 \\ 3 & 1 & 6 & 2 \\ 2 & 6 & 1 & 3 \\ 6 & 2 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.