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SageMath
E = EllipticCurve("n1")
E.isogeny_class()
Elliptic curves in class 12675.n
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 12675.n1 | 12675x7 | \([1, 0, 0, -9126088, -10612219333]\) | \(1114544804970241/405\) | \(30544650703125\) | \([2]\) | \(221184\) | \(2.3781\) | |
| 12675.n2 | 12675x5 | \([1, 0, 0, -570463, -165801208]\) | \(272223782641/164025\) | \(12370583534765625\) | \([2, 2]\) | \(110592\) | \(2.0315\) | |
| 12675.n3 | 12675x8 | \([1, 0, 0, -464838, -229070583]\) | \(-147281603041/215233605\) | \(-16232679714319453125\) | \([2]\) | \(221184\) | \(2.3781\) | |
| 12675.n4 | 12675x4 | \([1, 0, 0, -338088, 75636417]\) | \(56667352321/15\) | \(1131283359375\) | \([2]\) | \(55296\) | \(1.6849\) | |
| 12675.n5 | 12675x3 | \([1, 0, 0, -42338, -1554333]\) | \(111284641/50625\) | \(3818081337890625\) | \([2, 2]\) | \(55296\) | \(1.6849\) | |
| 12675.n6 | 12675x2 | \([1, 0, 0, -21213, 1170792]\) | \(13997521/225\) | \(16969250390625\) | \([2, 2]\) | \(27648\) | \(1.3383\) | |
| 12675.n7 | 12675x1 | \([1, 0, 0, -88, 51167]\) | \(-1/15\) | \(-1131283359375\) | \([2]\) | \(13824\) | \(0.99177\) | \(\Gamma_0(N)\)-optimal |
| 12675.n8 | 12675x6 | \([1, 0, 0, 147787, -11630958]\) | \(4733169839/3515625\) | \(-265144537353515625\) | \([2]\) | \(110592\) | \(2.0315\) |
Rank
sage: E.rank()
The elliptic curves in class 12675.n have rank \(0\).
Complex multiplication
The elliptic curves in class 12675.n do not have complex multiplication.Modular form 12675.2.a.n
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 16 & 4 & 8 & 16 & 8 \\ 2 & 1 & 2 & 8 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 16 & 4 & 8 & 16 & 8 \\ 16 & 8 & 16 & 1 & 4 & 2 & 4 & 8 \\ 4 & 2 & 4 & 4 & 1 & 2 & 4 & 2 \\ 8 & 4 & 8 & 2 & 2 & 1 & 2 & 4 \\ 16 & 8 & 16 & 4 & 4 & 2 & 1 & 8 \\ 8 & 4 & 8 & 8 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
