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SageMath

sage: E = EllipticCurve("b1")

sage: E.isogeny_class()

## Elliptic curves in class 464968b

sage: E.isogeny_class().curves

LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|

464968.b2 | 464968b1 | \([0, 1, 0, -10228, -515040]\) | \(-9826000/3703\) | \(-44597989719808\) | \([2]\) | \(1152000\) | \(1.3289\) |
\(\Gamma_0(N)\)-optimal^{*} |

464968.b1 | 464968b2 | \([0, 1, 0, -176288, -28545968]\) | \(12576878500/1127\) | \(54293204876288\) | \([2]\) | \(2304000\) | \(1.6755\) |
\(\Gamma_0(N)\)-optimal^{*} |

^{*}optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 2 curves highlighted, and conditionally curve 464968b1.

## Rank

sage: E.rank()

The elliptic curves in class 464968b have rank \(0\).

## Complex multiplication

The elliptic curves in class 464968b do not have complex multiplication.## Modular form 464968.2.a.b

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.

\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)

## Isogeny graph

sage: E.isogeny_graph().plot(edge_labels=True)

The vertices are labelled with Cremona labels.