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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 2310.d
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
2310.d1 | 2310d3 | \([1, 1, 0, -2137, 37129]\) | \(1080077156587801/594247500\) | \(594247500\) | \([4]\) | \(2048\) | \(0.63163\) | |
2310.d2 | 2310d2 | \([1, 1, 0, -157, 301]\) | \(432252699481/192099600\) | \(192099600\) | \([2, 2]\) | \(1024\) | \(0.28505\) | |
2310.d3 | 2310d1 | \([1, 1, 0, -77, -291]\) | \(51520374361/887040\) | \(887040\) | \([2]\) | \(512\) | \(-0.061519\) | \(\Gamma_0(N)\)-optimal |
2310.d4 | 2310d4 | \([1, 1, 0, 543, 2961]\) | \(17655210697319/13448344140\) | \(-13448344140\) | \([2]\) | \(2048\) | \(0.63163\) |
Rank
sage: E.rank()
The elliptic curves in class 2310.d have rank \(1\).
Complex multiplication
The elliptic curves in class 2310.d do not have complex multiplication.Modular form 2310.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.