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SageMath
E = EllipticCurve("bd1")
E.isogeny_class()
Elliptic curves in class 187425bd
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 187425.bb3 | 187425bd1 | \([1, -1, 1, -3693605, -2731338228]\) | \(4158523459441/16065\) | \(21528636466640625\) | \([2]\) | \(3538944\) | \(2.3475\) | \(\Gamma_0(N)\)-optimal |
| 187425.bb2 | 187425bd2 | \([1, -1, 1, -3748730, -2645563728]\) | \(4347507044161/258084225\) | \(345857544836581640625\) | \([2, 2]\) | \(7077888\) | \(2.6941\) | |
| 187425.bb1 | 187425bd3 | \([1, -1, 1, -11190605, 11121905022]\) | \(115650783909361/27072079335\) | \(36279175499487648984375\) | \([2]\) | \(14155776\) | \(3.0406\) | |
| 187425.bb4 | 187425bd4 | \([1, -1, 1, 2811145, -10924125978]\) | \(1833318007919/39525924375\) | \(-52968518946610927734375\) | \([2]\) | \(14155776\) | \(3.0406\) |
Rank
sage: E.rank()
The elliptic curves in class 187425bd have rank \(1\).
Complex multiplication
The elliptic curves in class 187425bd do not have complex multiplication.Modular form 187425.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
