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SageMath
E = EllipticCurve("eh1")
E.isogeny_class()
Elliptic curves in class 55440.eh
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 55440.eh1 | 55440en4 | \([0, 0, 0, -4750707, 3985475506]\) | \(3971101377248209009/56495958750\) | \(168696028892160000\) | \([2]\) | \(1179648\) | \(2.4452\) | |
| 55440.eh2 | 55440en2 | \([0, 0, 0, -305427, 58515154]\) | \(1055257664218129/115307784900\) | \(344307200786841600\) | \([2, 2]\) | \(589824\) | \(2.0987\) | |
| 55440.eh3 | 55440en1 | \([0, 0, 0, -72147, -6476654]\) | \(13908844989649/1980372240\) | \(5913359822684160\) | \([2]\) | \(294912\) | \(1.7521\) | \(\Gamma_0(N)\)-optimal |
| 55440.eh4 | 55440en3 | \([0, 0, 0, 407373, 291030514]\) | \(2503876820718671/13702874328990\) | \(-40916563500374876160\) | \([2]\) | \(1179648\) | \(2.4452\) |
Rank
sage: E.rank()
The elliptic curves in class 55440.eh have rank \(1\).
Complex multiplication
The elliptic curves in class 55440.eh do not have complex multiplication.Modular form 55440.2.a.eh
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
