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SageMath
E = EllipticCurve("eq1")
E.isogeny_class()
Elliptic curves in class 55440eq
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 55440.dy3 | 55440eq1 | \([0, 0, 0, -1587, -19406]\) | \(148035889/31185\) | \(93117911040\) | \([2]\) | \(49152\) | \(0.81945\) | \(\Gamma_0(N)\)-optimal |
| 55440.dy2 | 55440eq2 | \([0, 0, 0, -8067, 261826]\) | \(19443408769/1334025\) | \(3983377305600\) | \([2, 2]\) | \(98304\) | \(1.1660\) | |
| 55440.dy4 | 55440eq3 | \([0, 0, 0, 7053, 1129714]\) | \(12994449551/192163125\) | \(-573796016640000\) | \([2]\) | \(196608\) | \(1.5126\) | |
| 55440.dy1 | 55440eq4 | \([0, 0, 0, -126867, 17392786]\) | \(75627935783569/396165\) | \(1182942351360\) | \([4]\) | \(196608\) | \(1.5126\) |
Rank
sage: E.rank()
The elliptic curves in class 55440eq have rank \(1\).
Complex multiplication
The elliptic curves in class 55440eq do not have complex multiplication.Modular form 55440.2.a.eq
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
