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SageMath
E = EllipticCurve("bp1")
E.isogeny_class()
Elliptic curves in class 60648bp
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 60648.bp4 | 60648bp1 | \([0, 1, 0, -2647, -341038]\) | \(-2725888/64827\) | \(-48797493241392\) | \([2]\) | \(165888\) | \(1.3073\) | \(\Gamma_0(N)\)-optimal |
| 60648.bp3 | 60648bp2 | \([0, 1, 0, -91092, -10565280]\) | \(6940769488/35721\) | \(430214634291456\) | \([2, 2]\) | \(331776\) | \(1.6538\) | |
| 60648.bp2 | 60648bp3 | \([0, 1, 0, -141632, 2413392]\) | \(6522128932/3720087\) | \(179215124799126528\) | \([2]\) | \(663552\) | \(2.0004\) | |
| 60648.bp1 | 60648bp4 | \([0, 1, 0, -1455672, -676480320]\) | \(7080974546692/189\) | \(9105071625216\) | \([2]\) | \(663552\) | \(2.0004\) |
Rank
sage: E.rank()
The elliptic curves in class 60648bp have rank \(1\).
Complex multiplication
The elliptic curves in class 60648bp do not have complex multiplication.Modular form 60648.2.a.bp
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.
