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SageMath
E = EllipticCurve("bd1")
E.isogeny_class()
Elliptic curves in class 114240.bd
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
114240.bd1 | 114240fn4 | \([0, -1, 0, -64961, -4896159]\) | \(115650783909361/27072079335\) | \(7096783165194240\) | \([2]\) | \(786432\) | \(1.7534\) | |
114240.bd2 | 114240fn2 | \([0, -1, 0, -21761, 1177761]\) | \(4347507044161/258084225\) | \(67655231078400\) | \([2, 2]\) | \(393216\) | \(1.4068\) | |
114240.bd3 | 114240fn1 | \([0, -1, 0, -21441, 1215585]\) | \(4158523459441/16065\) | \(4211343360\) | \([2]\) | \(196608\) | \(1.0602\) | \(\Gamma_0(N)\)-optimal |
114240.bd4 | 114240fn3 | \([0, -1, 0, 16319, 4825825]\) | \(1833318007919/39525924375\) | \(-10361483919360000\) | \([2]\) | \(786432\) | \(1.7534\) |
Rank
sage: E.rank()
The elliptic curves in class 114240.bd have rank \(1\).
Complex multiplication
The elliptic curves in class 114240.bd do not have complex multiplication.Modular form 114240.2.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 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.