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SageMath
E = EllipticCurve("en1")
E.isogeny_class()
Elliptic curves in class 163170en
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 163170.u2 | 163170en1 | \([1, -1, 0, -575955, 253384501]\) | \(-246362173188769/180452966400\) | \(-15476750951071334400\) | \([2]\) | \(3440640\) | \(2.3816\) | \(\Gamma_0(N)\)-optimal |
| 163170.u1 | 163170en2 | \([1, -1, 0, -10454355, 13010350261]\) | \(1473328864410526369/362154602880\) | \(31060595491313028480\) | \([2]\) | \(6881280\) | \(2.7282\) |
Rank
sage: E.rank()
The elliptic curves in class 163170en have rank \(1\).
Complex multiplication
The elliptic curves in class 163170en do not have complex multiplication.Modular form 163170.2.a.en
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.
