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SageMath
E = EllipticCurve("l1")
E.isogeny_class()
Elliptic curves in class 212415l
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 212415.y5 | 212415l1 | \([1, 0, 0, 495340, -332077425]\) | \(4733169839/19518975\) | \(-55429221140208456975\) | \([4]\) | \(7077888\) | \(2.4711\) | \(\Gamma_0(N)\)-optimal |
| 212415.y4 | 212415l2 | \([1, 0, 0, -5239865, -4074872208]\) | \(5602762882081/716900625\) | \(2035826332001483450625\) | \([2, 2]\) | \(14155776\) | \(2.8177\) | |
| 212415.y3 | 212415l3 | \([1, 0, 0, -21170990, 33302733267]\) | \(369543396484081/45120132225\) | \(128130385277657809530225\) | \([2, 2]\) | \(28311552\) | \(3.1642\) | |
| 212415.y2 | 212415l4 | \([1, 0, 0, -81072020, -280968402975]\) | \(20751759537944401/418359375\) | \(1188040576564824609375\) | \([2]\) | \(28311552\) | \(3.1642\) | |
| 212415.y1 | 212415l5 | \([1, 0, 0, -328110665, 2287529094402]\) | \(1375634265228629281/24990412335\) | \(70966794706264452291135\) | \([2]\) | \(56623104\) | \(3.5108\) | |
| 212415.y6 | 212415l6 | \([1, 0, 0, 30870685, 171306847032]\) | \(1145725929069119/5127181719135\) | \(-14559969943911915523501935\) | \([2]\) | \(56623104\) | \(3.5108\) |
Rank
sage: E.rank()
The elliptic curves in class 212415l have rank \(2\).
Complex multiplication
The elliptic curves in class 212415l do not have complex multiplication.Modular form 212415.2.a.l
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.
