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SageMath
E = EllipticCurve("dr1")
E.isogeny_class()
Elliptic curves in class 134064.dr
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 134064.dr1 | 134064dd3 | \([0, 0, 0, -119314755, 357541668162]\) | \(19804628171203875/5638671302656\) | \(53483061497922343604846592\) | \([2]\) | \(31850496\) | \(3.6435\) | |
| 134064.dr2 | 134064dd1 | \([0, 0, 0, -109518675, 441143988754]\) | \(11165451838341046875/572244736\) | \(7445498124422873088\) | \([2]\) | \(10616832\) | \(3.0941\) | \(\Gamma_0(N)\)-optimal |
| 134064.dr3 | 134064dd2 | \([0, 0, 0, -109330515, 442735333138]\) | \(-11108001800138902875/79947274872976\) | \(-1040197047997769082667008\) | \([2]\) | \(21233664\) | \(3.4407\) | |
| 134064.dr4 | 134064dd4 | \([0, 0, 0, 314205885, 2358586238274]\) | \(361682234074684125/462672528510976\) | \(-4388470610814994722777464832\) | \([2]\) | \(63700992\) | \(3.9900\) |
Rank
sage: E.rank()
The elliptic curves in class 134064.dr have rank \(0\).
Complex multiplication
The elliptic curves in class 134064.dr do not have complex multiplication.Modular form 134064.2.a.dr
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 LMFDB numbering.
\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
