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SageMath
E = EllipticCurve("h1")
E.isogeny_class()
Elliptic curves in class 8925h
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 8925.w5 | 8925h1 | \([1, 1, 0, 875, 25000]\) | \(4733169839/19518975\) | \(-304983984375\) | \([2]\) | \(12288\) | \(0.88625\) | \(\Gamma_0(N)\)-optimal |
| 8925.w4 | 8925h2 | \([1, 1, 0, -9250, 298375]\) | \(5602762882081/716900625\) | \(11201572265625\) | \([2, 2]\) | \(24576\) | \(1.2328\) | |
| 8925.w3 | 8925h3 | \([1, 1, 0, -37375, -2486000]\) | \(369543396484081/45120132225\) | \(705002066015625\) | \([2, 2]\) | \(49152\) | \(1.5794\) | |
| 8925.w2 | 8925h4 | \([1, 1, 0, -143125, 20781250]\) | \(20751759537944401/418359375\) | \(6536865234375\) | \([2]\) | \(49152\) | \(1.5794\) | |
| 8925.w1 | 8925h5 | \([1, 1, 0, -579250, -169925375]\) | \(1375634265228629281/24990412335\) | \(390475192734375\) | \([2]\) | \(98304\) | \(1.9260\) | |
| 8925.w6 | 8925h6 | \([1, 1, 0, 54500, -12684125]\) | \(1145725929069119/5127181719135\) | \(-80112214361484375\) | \([2]\) | \(98304\) | \(1.9260\) |
Rank
sage: E.rank()
The elliptic curves in class 8925h have rank \(0\).
Complex multiplication
The elliptic curves in class 8925h do not have complex multiplication.Modular form 8925.2.a.h
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
