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SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 40425.i
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 40425.i1 | 40425m4 | \([1, 1, 1, -25360588, 13916648906]\) | \(981281029968144361/522287841796875\) | \(960103785930633544921875\) | \([2]\) | \(5308416\) | \(3.2932\) | |
| 40425.i2 | 40425m2 | \([1, 1, 1, -19903213, 34130765906]\) | \(474334834335054841/607815140625\) | \(1117325679365478515625\) | \([2, 2]\) | \(2654208\) | \(2.9466\) | |
| 40425.i3 | 40425m1 | \([1, 1, 1, -19897088, 34152852656]\) | \(473897054735271721/779625\) | \(1433157837890625\) | \([4]\) | \(1327104\) | \(2.6000\) | \(\Gamma_0(N)\)-optimal |
| 40425.i4 | 40425m3 | \([1, 1, 1, -14543838, 52931453406]\) | \(-185077034913624841/551466161890875\) | \(-1013741288754680513671875\) | \([2]\) | \(5308416\) | \(3.2932\) |
Rank
sage: E.rank()
The elliptic curves in class 40425.i have rank \(0\).
Complex multiplication
The elliptic curves in class 40425.i do not have complex multiplication.Modular form 40425.2.a.i
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.
