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SageMath
E = EllipticCurve("bj1")
E.isogeny_class()
Elliptic curves in class 5040bj
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 5040.ba7 | 5040bj1 | \([0, 0, 0, -5907, -61486]\) | \(7633736209/3870720\) | \(11557907988480\) | \([2]\) | \(9216\) | \(1.1990\) | \(\Gamma_0(N)\)-optimal |
| 5040.ba5 | 5040bj2 | \([0, 0, 0, -51987, 4518866]\) | \(5203798902289/57153600\) | \(170659735142400\) | \([2, 2]\) | \(18432\) | \(1.5456\) | |
| 5040.ba4 | 5040bj3 | \([0, 0, 0, -386067, -92329774]\) | \(2131200347946769/2058000\) | \(6145155072000\) | \([2]\) | \(27648\) | \(1.7483\) | |
| 5040.ba2 | 5040bj4 | \([0, 0, 0, -829587, 290831186]\) | \(21145699168383889/2593080\) | \(7742895390720\) | \([2]\) | \(36864\) | \(1.8922\) | |
| 5040.ba6 | 5040bj5 | \([0, 0, 0, -11667, 11349074]\) | \(-58818484369/18600435000\) | \(-55540601303040000\) | \([2]\) | \(36864\) | \(1.8922\) | |
| 5040.ba3 | 5040bj6 | \([0, 0, 0, -388947, -90882286]\) | \(2179252305146449/66177562500\) | \(197605142784000000\) | \([2, 2]\) | \(55296\) | \(2.0949\) | |
| 5040.ba1 | 5040bj7 | \([0, 0, 0, -928947, 216809714]\) | \(29689921233686449/10380965400750\) | \(30997396591193088000\) | \([2]\) | \(110592\) | \(2.4415\) | |
| 5040.ba8 | 5040bj8 | \([0, 0, 0, 104973, -305935054]\) | \(42841933504271/13565917968750\) | \(-40507614000000000000\) | \([2]\) | \(110592\) | \(2.4415\) |
Rank
sage: E.rank()
The elliptic curves in class 5040bj have rank \(0\).
Complex multiplication
The elliptic curves in class 5040bj do not have complex multiplication.Modular form 5040.2.a.bj
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
