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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 3600.e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality | CM discriminant |
---|---|---|---|---|---|---|---|---|---|
3600.e1 | 3600bb4 | \([0, 0, 0, -3375, 74250]\) | \(54000\) | \(78732000000\) | \([2]\) | \(3456\) | \(0.88518\) | \(-12\) | |
3600.e2 | 3600bb2 | \([0, 0, 0, -375, -2750]\) | \(54000\) | \(108000000\) | \([2]\) | \(1152\) | \(0.33588\) | \(-12\) | |
3600.e3 | 3600bb1 | \([0, 0, 0, 0, -125]\) | \(0\) | \(-6750000\) | \([2]\) | \(576\) | \(-0.010696\) | \(\Gamma_0(N)\)-optimal | \(-3\) |
3600.e4 | 3600bb3 | \([0, 0, 0, 0, 3375]\) | \(0\) | \(-4920750000\) | \([2]\) | \(1728\) | \(0.53861\) | \(-3\) |
Rank
sage: E.rank()
The elliptic curves in class 3600.e have rank \(0\).
Complex multiplication
Each elliptic curve in class 3600.e has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-3}) \).Modular form 3600.2.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.
\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.