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SageMath
sage: E = EllipticCurve("by1")
sage: E.isogeny_class()
Elliptic curves in class 28224by
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 28224.fl5 | 28224by1 | \([0, 0, 0, 1176, 19208]\) | \(2048/3\) | \(-263473523712\) | \([2]\) | \(24576\) | \(0.87691\) | \(\Gamma_0(N)\)-optimal |
| 28224.fl4 | 28224by2 | \([0, 0, 0, -7644, 192080]\) | \(35152/9\) | \(12646729138176\) | \([2, 2]\) | \(49152\) | \(1.2235\) | |
| 28224.fl3 | 28224by3 | \([0, 0, 0, -42924, -3265360]\) | \(1556068/81\) | \(455282248974336\) | \([2, 2]\) | \(98304\) | \(1.5701\) | |
| 28224.fl2 | 28224by4 | \([0, 0, 0, -113484, 14713328]\) | \(28756228/3\) | \(16862305517568\) | \([2]\) | \(98304\) | \(1.5701\) | |
| 28224.fl6 | 28224by5 | \([0, 0, 0, 27636, -12946192]\) | \(207646/6561\) | \(-73755724333842432\) | \([2]\) | \(196608\) | \(1.9166\) | |
| 28224.fl1 | 28224by6 | \([0, 0, 0, -677964, -214860688]\) | \(3065617154/9\) | \(101173833105408\) | \([2]\) | \(196608\) | \(1.9166\) |
Rank
sage: E.rank()
The elliptic curves in class 28224by have rank \(1\).
Complex multiplication
The elliptic curves in class 28224by do not have complex multiplication.Modular form 28224.2.a.by
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
