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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 221952.d
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 221952.d1 | 221952bh3 | \([0, -1, 0, -187079, -31082421]\) | \(58591911104/243\) | \(3003099784704\) | \([2]\) | \(819200\) | \(1.6036\) | |
| 221952.d2 | 221952bh4 | \([0, -1, 0, -184189, -32092187]\) | \(-873722816/59049\) | \(-46704207851716608\) | \([2]\) | \(1638400\) | \(1.9502\) | |
| 221952.d3 | 221952bh1 | \([0, -1, 0, -2119, 37099]\) | \(85184/3\) | \(37075305984\) | \([2]\) | \(163840\) | \(0.79889\) | \(\Gamma_0(N)\)-optimal |
| 221952.d4 | 221952bh2 | \([0, -1, 0, 771, 127845]\) | \(64/9\) | \(-7118458748928\) | \([2]\) | \(327680\) | \(1.1455\) |
Rank
sage: E.rank()
The elliptic curves in class 221952.d have rank \(1\).
Complex multiplication
The elliptic curves in class 221952.d do not have complex multiplication.Modular form 221952.2.a.d
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 5 & 10 \\ 2 & 1 & 10 & 5 \\ 5 & 10 & 1 & 2 \\ 10 & 5 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
