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SageMath
E = EllipticCurve("fz1")
E.isogeny_class()
Elliptic curves in class 163170fz
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 163170.t1 | 163170fz1 | \([1, -1, 0, -494370, -133663804]\) | \(12264063975981/370000\) | \(403132533930000\) | \([2]\) | \(1376256\) | \(1.9008\) | \(\Gamma_0(N)\)-optimal |
| 163170.t2 | 163170fz2 | \([1, -1, 0, -473790, -145316200]\) | \(-10795326296301/2139062500\) | \(-2330609961782812500\) | \([2]\) | \(2752512\) | \(2.2474\) |
Rank
sage: E.rank()
The elliptic curves in class 163170fz have rank \(2\).
Complex multiplication
The elliptic curves in class 163170fz do not have complex multiplication.Modular form 163170.2.a.fz
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.
