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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 2415.d
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
2415.d1 | 2415i1 | \([1, 0, 0, -230, -1173]\) | \(1345938541921/203765625\) | \(203765625\) | \([2]\) | \(768\) | \(0.31835\) | \(\Gamma_0(N)\)-optimal |
2415.d2 | 2415i2 | \([1, 0, 0, 395, -6298]\) | \(6814692748079/21258460125\) | \(-21258460125\) | \([2]\) | \(1536\) | \(0.66492\) |
Rank
sage: E.rank()
The elliptic curves in class 2415.d have rank \(1\).
Complex multiplication
The elliptic curves in class 2415.d do not have complex multiplication.Modular form 2415.2.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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.