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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 51714p
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 51714.r5 | 51714p1 | \([1, -1, 1, -51746, -4188895]\) | \(4354703137/352512\) | \(1240399400677632\) | \([2]\) | \(294912\) | \(1.6391\) | \(\Gamma_0(N)\)-optimal |
| 51714.r4 | 51714p2 | \([1, -1, 1, -173426, 22970081]\) | \(163936758817/30338064\) | \(106751873420818704\) | \([2, 2]\) | \(589824\) | \(1.9857\) | |
| 51714.r6 | 51714p3 | \([1, -1, 1, 343714, 133431185]\) | \(1276229915423/2927177028\) | \(-10299985904617522308\) | \([2]\) | \(1179648\) | \(2.3323\) | |
| 51714.r2 | 51714p4 | \([1, -1, 1, -2637446, 1649223281]\) | \(576615941610337/27060804\) | \(95220035242643844\) | \([2, 2]\) | \(1179648\) | \(2.3323\) | |
| 51714.r3 | 51714p5 | \([1, -1, 1, -2500556, 1827946865]\) | \(-491411892194497/125563633938\) | \(-441826253527825360818\) | \([2]\) | \(2359296\) | \(2.6789\) | |
| 51714.r1 | 51714p6 | \([1, -1, 1, -42198656, 105521136257]\) | \(2361739090258884097/5202\) | \(18304505044722\) | \([2]\) | \(2359296\) | \(2.6789\) |
Rank
sage: E.rank()
The elliptic curves in class 51714p have rank \(1\).
Complex multiplication
The elliptic curves in class 51714p do not have complex multiplication.Modular form 51714.2.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 Cremona numbering.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
