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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 17325.p
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 17325.p1 | 17325y5 | \([1, -1, 1, -2981255, 1981934372]\) | \(257260669489908001/14267882475\) | \(162520098816796875\) | \([2]\) | \(393216\) | \(2.3679\) | |
| 17325.p2 | 17325y3 | \([1, -1, 1, -196880, 27303122]\) | \(74093292126001/14707625625\) | \(167529048134765625\) | \([2, 2]\) | \(196608\) | \(2.0213\) | |
| 17325.p3 | 17325y2 | \([1, -1, 1, -60755, -5366878]\) | \(2177286259681/161417025\) | \(1838640800390625\) | \([2, 2]\) | \(98304\) | \(1.6747\) | |
| 17325.p4 | 17325y1 | \([1, -1, 1, -59630, -5589628]\) | \(2058561081361/12705\) | \(144717890625\) | \([2]\) | \(49152\) | \(1.3282\) | \(\Gamma_0(N)\)-optimal |
| 17325.p5 | 17325y4 | \([1, -1, 1, 57370, -23794378]\) | \(1833318007919/22507682505\) | \(-256376571033515625\) | \([2]\) | \(196608\) | \(2.0213\) | |
| 17325.p6 | 17325y6 | \([1, -1, 1, 409495, 161918372]\) | \(666688497209279/1381398046875\) | \(-15734987127685546875\) | \([2]\) | \(393216\) | \(2.3679\) |
Rank
sage: E.rank()
The elliptic curves in class 17325.p have rank \(1\).
Complex multiplication
The elliptic curves in class 17325.p do not have complex multiplication.Modular form 17325.2.a.p
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 8 & 4 \\ 2 & 1 & 2 & 4 & 4 & 2 \\ 4 & 2 & 1 & 2 & 2 & 4 \\ 8 & 4 & 2 & 1 & 4 & 8 \\ 8 & 4 & 2 & 4 & 1 & 8 \\ 4 & 2 & 4 & 8 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
