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SageMath
E = EllipticCurve("ge1")
E.isogeny_class()
Elliptic curves in class 47040ge
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 47040.fj3 | 47040ge1 | \([0, 1, 0, -259716, -47586330]\) | \(257307998572864/19456203375\) | \(146496183735384000\) | \([2]\) | \(442368\) | \(2.0389\) | \(\Gamma_0(N)\)-optimal |
| 47040.fj2 | 47040ge2 | \([0, 1, 0, -847961, 244300839]\) | \(139927692143296/27348890625\) | \(13179165217344000000\) | \([2, 2]\) | \(884736\) | \(2.3855\) | |
| 47040.fj4 | 47040ge3 | \([0, 1, 0, 1745119, 1448008575]\) | \(152461584507448/322998046875\) | \(-1245197016000000000000\) | \([2]\) | \(1769472\) | \(2.7321\) | |
| 47040.fj1 | 47040ge4 | \([0, 1, 0, -12852961, 17730783839]\) | \(60910917333827912/3255076125\) | \(12548716987355136000\) | \([2]\) | \(1769472\) | \(2.7321\) |
Rank
sage: E.rank()
The elliptic curves in class 47040ge have rank \(0\).
Complex multiplication
The elliptic curves in class 47040ge do not have complex multiplication.Modular form 47040.2.a.ge
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.
