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SageMath
E = EllipticCurve("bd1")
E.isogeny_class()
Elliptic curves in class 422370bd
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
422370.bd1 | 422370bd1 | \([1, -1, 0, -111075, -14584259]\) | \(-119313478467/3566680\) | \(-4530535276817160\) | \([]\) | \(3525120\) | \(1.7823\) | \(\Gamma_0(N)\)-optimal* |
422370.bd2 | 422370bd2 | \([1, -1, 0, 506235, -56245825]\) | \(15494117157/10435750\) | \(-9663547033226297250\) | \([]\) | \(10575360\) | \(2.3316\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 422370bd have rank \(0\).
Complex multiplication
The elliptic curves in class 422370bd do not have complex multiplication.Modular form 422370.2.a.bd
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 Cremona numbering.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.