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SageMath
E = EllipticCurve("w1")
E.isogeny_class()
Elliptic curves in class 17424.w
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality | CM discriminant |
---|---|---|---|---|---|---|---|---|---|
17424.w1 | 17424be4 | \([0, 0, 0, -16335, -790614]\) | \(54000\) | \(8926626601728\) | \([2]\) | \(34560\) | \(1.2794\) | \(-12\) | |
17424.w2 | 17424be2 | \([0, 0, 0, -1815, 29282]\) | \(54000\) | \(12245029632\) | \([2]\) | \(11520\) | \(0.73011\) | \(-12\) | |
17424.w3 | 17424be3 | \([0, 0, 0, 0, -35937]\) | \(0\) | \(-557914162608\) | \([2]\) | \(17280\) | \(0.93284\) | \(-3\) | |
17424.w4 | 17424be1 | \([0, 0, 0, 0, 1331]\) | \(0\) | \(-765314352\) | \([2]\) | \(5760\) | \(0.38353\) | \(\Gamma_0(N)\)-optimal | \(-3\) |
Rank
sage: E.rank()
The elliptic curves in class 17424.w have rank \(1\).
Complex multiplication
Each elliptic curve in class 17424.w has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-3}) \).Modular form 17424.2.a.w
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.