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SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 135424j
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality | CM discriminant |
---|---|---|---|---|---|---|---|---|---|
135424.t2 | 135424j1 | \([0, -1, 0, -1763, 25823]\) | \(8000\) | \(75794375168\) | \([2]\) | \(101376\) | \(0.81720\) | \(\Gamma_0(N)\)-optimal | \(-8\) |
135424.t1 | 135424j2 | \([0, -1, 0, -7053, -199531]\) | \(8000\) | \(4850840010752\) | \([2]\) | \(202752\) | \(1.1638\) | \(-8\) |
Rank
sage: E.rank()
The elliptic curves in class 135424j have rank \(1\).
Complex multiplication
Each elliptic curve in class 135424j has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-2}) \).Modular form 135424.2.a.j
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.