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SageMath
E = EllipticCurve("g1")
E.isogeny_class()
Elliptic curves in class 1040.g
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 1040.g1 | 1040g3 | \([0, -1, 0, -3320, 71792]\) | \(988345570681/44994560\) | \(184297717760\) | \([2]\) | \(1728\) | \(0.92360\) | |
| 1040.g2 | 1040g1 | \([0, -1, 0, -520, -4368]\) | \(3803721481/26000\) | \(106496000\) | \([2]\) | \(576\) | \(0.37429\) | \(\Gamma_0(N)\)-optimal |
| 1040.g3 | 1040g2 | \([0, -1, 0, -200, -10000]\) | \(-217081801/10562500\) | \(-43264000000\) | \([2]\) | \(1152\) | \(0.72086\) | |
| 1040.g4 | 1040g4 | \([0, -1, 0, 1800, 268400]\) | \(157376536199/7722894400\) | \(-31632975462400\) | \([2]\) | \(3456\) | \(1.2702\) |
Rank
sage: E.rank()
The elliptic curves in class 1040.g have rank \(0\).
Complex multiplication
The elliptic curves in class 1040.g do not have complex multiplication.Modular form 1040.2.a.g
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.
\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
