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SageMath
E = EllipticCurve("bz1")
E.isogeny_class()
Elliptic curves in class 15600bz
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 15600.cj6 | 15600bz1 | \([0, 1, 0, -44008, -3568012]\) | \(147281603041/5265\) | \(336960000000\) | \([2]\) | \(36864\) | \(1.3010\) | \(\Gamma_0(N)\)-optimal |
| 15600.cj5 | 15600bz2 | \([0, 1, 0, -46008, -3228012]\) | \(168288035761/27720225\) | \(1774094400000000\) | \([2, 2]\) | \(73728\) | \(1.6476\) | |
| 15600.cj4 | 15600bz3 | \([0, 1, 0, -208008, 33383988]\) | \(15551989015681/1445900625\) | \(92537640000000000\) | \([2, 2]\) | \(147456\) | \(1.9941\) | |
| 15600.cj7 | 15600bz4 | \([0, 1, 0, 83992, -18048012]\) | \(1023887723039/2798036865\) | \(-179074359360000000\) | \([2]\) | \(147456\) | \(1.9941\) | |
| 15600.cj2 | 15600bz5 | \([0, 1, 0, -3250008, 2254043988]\) | \(59319456301170001/594140625\) | \(38025000000000000\) | \([2, 2]\) | \(294912\) | \(2.3407\) | |
| 15600.cj8 | 15600bz6 | \([0, 1, 0, 241992, 158483988]\) | \(24487529386319/183539412225\) | \(-11746522382400000000\) | \([2]\) | \(294912\) | \(2.3407\) | |
| 15600.cj1 | 15600bz7 | \([0, 1, 0, -52000008, 144311543988]\) | \(242970740812818720001/24375\) | \(1560000000000\) | \([2]\) | \(589824\) | \(2.6873\) | |
| 15600.cj3 | 15600bz8 | \([0, 1, 0, -3172008, 2367455988]\) | \(-55150149867714721/5950927734375\) | \(-380859375000000000000\) | \([2]\) | \(589824\) | \(2.6873\) |
Rank
sage: E.rank()
The elliptic curves in class 15600bz have rank \(1\).
Complex multiplication
The elliptic curves in class 15600bz do not have complex multiplication.Modular form 15600.2.a.bz
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 8 & 16 & 16 \\ 2 & 1 & 2 & 2 & 4 & 4 & 8 & 8 \\ 4 & 2 & 1 & 4 & 2 & 2 & 4 & 4 \\ 4 & 2 & 4 & 1 & 8 & 8 & 16 & 16 \\ 8 & 4 & 2 & 8 & 1 & 4 & 2 & 2 \\ 8 & 4 & 2 & 8 & 4 & 1 & 8 & 8 \\ 16 & 8 & 4 & 16 & 2 & 8 & 1 & 4 \\ 16 & 8 & 4 & 16 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
