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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 6240.d
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
6240.d1 | 6240a3 | \([0, -1, 0, -1041, 13281]\) | \(30488290624/195\) | \(798720\) | \([2]\) | \(1536\) | \(0.31720\) | |
6240.d2 | 6240a2 | \([0, -1, 0, -216, -924]\) | \(2186875592/428415\) | \(219348480\) | \([2]\) | \(1536\) | \(0.31720\) | |
6240.d3 | 6240a1 | \([0, -1, 0, -66, 216]\) | \(504358336/38025\) | \(2433600\) | \([2, 2]\) | \(768\) | \(-0.029371\) | \(\Gamma_0(N)\)-optimal |
6240.d4 | 6240a4 | \([0, -1, 0, 64, 840]\) | \(55742968/658125\) | \(-336960000\) | \([2]\) | \(1536\) | \(0.31720\) |
Rank
sage: E.rank()
The elliptic curves in class 6240.d have rank \(1\).
Complex multiplication
The elliptic curves in class 6240.d do not have complex multiplication.Modular form 6240.2.a.d
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.