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SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 2535.j
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
2535.j1 | 2535a7 | \([1, 1, 0, -365043, -85043772]\) | \(1114544804970241/405\) | \(1954857645\) | \([2]\) | \(9216\) | \(1.5733\) | |
2535.j2 | 2535a5 | \([1, 1, 0, -22818, -1335537]\) | \(272223782641/164025\) | \(791717346225\) | \([2, 2]\) | \(4608\) | \(1.2268\) | |
2535.j3 | 2535a8 | \([1, 1, 0, -18593, -1840002]\) | \(-147281603041/215233605\) | \(-1038891501716445\) | \([2]\) | \(9216\) | \(1.5733\) | |
2535.j4 | 2535a4 | \([1, 1, 0, -13523, 599682]\) | \(56667352321/15\) | \(72402135\) | \([2]\) | \(2304\) | \(0.88020\) | |
2535.j5 | 2535a3 | \([1, 1, 0, -1693, -13112]\) | \(111284641/50625\) | \(244357205625\) | \([2, 2]\) | \(2304\) | \(0.88020\) | |
2535.j6 | 2535a2 | \([1, 1, 0, -848, 9027]\) | \(13997521/225\) | \(1086032025\) | \([2, 2]\) | \(1152\) | \(0.53362\) | |
2535.j7 | 2535a1 | \([1, 1, 0, -3, 408]\) | \(-1/15\) | \(-72402135\) | \([2]\) | \(576\) | \(0.18705\) | \(\Gamma_0(N)\)-optimal |
2535.j8 | 2535a6 | \([1, 1, 0, 5912, -90683]\) | \(4733169839/3515625\) | \(-16969250390625\) | \([2]\) | \(4608\) | \(1.2268\) |
Rank
sage: E.rank()
The elliptic curves in class 2535.j have rank \(1\).
Complex multiplication
The elliptic curves in class 2535.j do not have complex multiplication.Modular form 2535.2.a.j
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 16 & 4 & 8 & 16 & 8 \\ 2 & 1 & 2 & 8 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 16 & 4 & 8 & 16 & 8 \\ 16 & 8 & 16 & 1 & 4 & 2 & 4 & 8 \\ 4 & 2 & 4 & 4 & 1 & 2 & 4 & 2 \\ 8 & 4 & 8 & 2 & 2 & 1 & 2 & 4 \\ 16 & 8 & 16 & 4 & 4 & 2 & 1 & 8 \\ 8 & 4 & 8 & 8 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.