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SageMath
E = EllipticCurve("t1")
E.isogeny_class()
Elliptic curves in class 82810t
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 82810.r1 | 82810t1 | \([1, -1, 0, -477710, 115533536]\) | \(9663597/980\) | \(1222656571799395460\) | \([2]\) | \(1437696\) | \(2.2065\) | \(\Gamma_0(N)\)-optimal |
| 82810.r2 | 82810t2 | \([1, -1, 0, 598820, 561001650]\) | \(19034163/120050\) | \(-149775430045425943850\) | \([2]\) | \(2875392\) | \(2.5531\) |
Rank
sage: E.rank()
The elliptic curves in class 82810t have rank \(1\).
Complex multiplication
The elliptic curves in class 82810t do not have complex multiplication.Modular form 82810.2.a.t
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.
