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SageMath
E = EllipticCurve("fd1")
E.isogeny_class()
Elliptic curves in class 48960fd
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 48960.en6 | 48960fd1 | \([0, 0, 0, -46092, 4679696]\) | \(-56667352321/16711680\) | \(-3193651781959680\) | \([2]\) | \(196608\) | \(1.6895\) | \(\Gamma_0(N)\)-optimal |
| 48960.en5 | 48960fd2 | \([0, 0, 0, -783372, 266856464]\) | \(278202094583041/16646400\) | \(3181176579686400\) | \([2, 2]\) | \(393216\) | \(2.0360\) | |
| 48960.en4 | 48960fd3 | \([0, 0, 0, -829452, 233697296]\) | \(330240275458561/67652010000\) | \(12928500443381760000\) | \([2, 2]\) | \(786432\) | \(2.3826\) | |
| 48960.en2 | 48960fd4 | \([0, 0, 0, -12533772, 17079328784]\) | \(1139466686381936641/4080\) | \(779700142080\) | \([2]\) | \(786432\) | \(2.3826\) | |
| 48960.en7 | 48960fd5 | \([0, 0, 0, 1762548, 1402170896]\) | \(3168685387909439/6278181696900\) | \(-1199779206146319974400\) | \([2]\) | \(1572864\) | \(2.7292\) | |
| 48960.en3 | 48960fd6 | \([0, 0, 0, -4158732, -3056963056]\) | \(41623544884956481/2962701562500\) | \(566181085593600000000\) | \([2, 2]\) | \(1572864\) | \(2.7292\) | |
| 48960.en8 | 48960fd7 | \([0, 0, 0, 3772788, -13339385584]\) | \(31077313442863199/420227050781250\) | \(-80306640000000000000000\) | \([2]\) | \(3145728\) | \(3.0758\) | |
| 48960.en1 | 48960fd8 | \([0, 0, 0, -65358732, -203376803056]\) | \(161572377633716256481/914742821250\) | \(174810075415511040000\) | \([2]\) | \(3145728\) | \(3.0758\) |
Rank
sage: E.rank()
The elliptic curves in class 48960fd have rank \(0\).
Complex multiplication
The elliptic curves in class 48960fd do not have complex multiplication.Modular form 48960.2.a.fd
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 8 & 16 & 16 \\ 2 & 1 & 2 & 2 & 4 & 4 & 8 & 8 \\ 4 & 2 & 1 & 4 & 2 & 2 & 4 & 4 \\ 4 & 2 & 4 & 1 & 8 & 8 & 16 & 16 \\ 8 & 4 & 2 & 8 & 1 & 4 & 8 & 8 \\ 8 & 4 & 2 & 8 & 4 & 1 & 2 & 2 \\ 16 & 8 & 4 & 16 & 8 & 2 & 1 & 4 \\ 16 & 8 & 4 & 16 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
