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SageMath
E = EllipticCurve("et1")
E.isogeny_class()
Elliptic curves in class 48960.et
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
48960.et1 | 48960cl8 | \([0, 0, 0, -65358732, 203376803056]\) | \(161572377633716256481/914742821250\) | \(174810075415511040000\) | \([2]\) | \(3145728\) | \(3.0758\) | |
48960.et2 | 48960cl4 | \([0, 0, 0, -12533772, -17079328784]\) | \(1139466686381936641/4080\) | \(779700142080\) | \([2]\) | \(786432\) | \(2.3826\) | |
48960.et3 | 48960cl6 | \([0, 0, 0, -4158732, 3056963056]\) | \(41623544884956481/2962701562500\) | \(566181085593600000000\) | \([2, 2]\) | \(1572864\) | \(2.7292\) | |
48960.et4 | 48960cl3 | \([0, 0, 0, -829452, -233697296]\) | \(330240275458561/67652010000\) | \(12928500443381760000\) | \([2, 2]\) | \(786432\) | \(2.3826\) | |
48960.et5 | 48960cl2 | \([0, 0, 0, -783372, -266856464]\) | \(278202094583041/16646400\) | \(3181176579686400\) | \([2, 2]\) | \(393216\) | \(2.0360\) | |
48960.et6 | 48960cl1 | \([0, 0, 0, -46092, -4679696]\) | \(-56667352321/16711680\) | \(-3193651781959680\) | \([2]\) | \(196608\) | \(1.6895\) | \(\Gamma_0(N)\)-optimal |
48960.et7 | 48960cl5 | \([0, 0, 0, 1762548, -1402170896]\) | \(3168685387909439/6278181696900\) | \(-1199779206146319974400\) | \([2]\) | \(1572864\) | \(2.7292\) | |
48960.et8 | 48960cl7 | \([0, 0, 0, 3772788, 13339385584]\) | \(31077313442863199/420227050781250\) | \(-80306640000000000000000\) | \([2]\) | \(3145728\) | \(3.0758\) |
Rank
sage: E.rank()
The elliptic curves in class 48960.et have rank \(1\).
Complex multiplication
The elliptic curves in class 48960.et do not have complex multiplication.Modular form 48960.2.a.et
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 LMFDB numbering.
\(\left(\begin{array}{rrrrrrrr} 1 & 16 & 2 & 4 & 8 & 16 & 8 & 4 \\ 16 & 1 & 8 & 4 & 2 & 4 & 8 & 16 \\ 2 & 8 & 1 & 2 & 4 & 8 & 4 & 2 \\ 4 & 4 & 2 & 1 & 2 & 4 & 2 & 4 \\ 8 & 2 & 4 & 2 & 1 & 2 & 4 & 8 \\ 16 & 4 & 8 & 4 & 2 & 1 & 8 & 16 \\ 8 & 8 & 4 & 2 & 4 & 8 & 1 & 8 \\ 4 & 16 & 2 & 4 & 8 & 16 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.