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SageMath
E = EllipticCurve("v1")
E.isogeny_class()
Elliptic curves in class 7744.v
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality | CM discriminant |
---|---|---|---|---|---|---|---|---|---|
7744.v1 | 7744t3 | \([0, 0, 0, -5324, -149072]\) | \(287496\) | \(58050510848\) | \([2]\) | \(5760\) | \(0.92814\) | \(-16\) | |
7744.v2 | 7744t4 | \([0, 0, 0, -5324, 149072]\) | \(287496\) | \(58050510848\) | \([2]\) | \(5760\) | \(0.92814\) | \(-16\) | |
7744.v3 | 7744t2 | \([0, 0, 0, -484, 0]\) | \(1728\) | \(7256313856\) | \([2, 2]\) | \(2880\) | \(0.58156\) | \(-4\) | |
7744.v4 | 7744t1 | \([0, 0, 0, 121, 0]\) | \(1728\) | \(-113379904\) | \([2]\) | \(1440\) | \(0.23499\) | \(\Gamma_0(N)\)-optimal | \(-4\) |
Rank
sage: E.rank()
The elliptic curves in class 7744.v have rank \(1\).
Complex multiplication
Each elliptic curve in class 7744.v has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-1}) \).Modular form 7744.2.a.v
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.