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SageMath
sage: E = EllipticCurve("d1")
sage: E.isogeny_class()
Elliptic curves in class 3025.d
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality | CM discriminant |
---|---|---|---|---|---|---|---|---|---|
3025.d1 | 3025a2 | \([0, 1, 1, -22183, -1312206]\) | \(-32768\) | \(-36842932671875\) | \([]\) | \(4752\) | \(1.3806\) | \(-11\) | |
3025.d2 | 3025a1 | \([0, 1, 1, -183, 919]\) | \(-32768\) | \(-20796875\) | \([]\) | \(432\) | \(0.18165\) | \(\Gamma_0(N)\)-optimal | \(-11\) |
Rank
sage: E.rank()
The elliptic curves in class 3025.d have rank \(1\).
Complex multiplication
Each elliptic curve in class 3025.d has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-11}) \).Modular form 3025.2.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.
\(\left(\begin{array}{rr} 1 & 11 \\ 11 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.