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SageMath
E = EllipticCurve("ei1")
E.isogeny_class()
Elliptic curves in class 58800.ei
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 58800.ei1 | 58800fz8 | \([0, -1, 0, -37647680408, -2811599839436688]\) | \(783736670177727068275201/360150\) | \(2711762390400000000\) | \([2]\) | \(56623104\) | \(4.2653\) | |
| 58800.ei2 | 58800fz6 | \([0, -1, 0, -2352980408, -43930644236688]\) | \(191342053882402567201/129708022500\) | \(976641224902560000000000\) | \([2, 2]\) | \(28311552\) | \(3.9187\) | |
| 58800.ei3 | 58800fz7 | \([0, -1, 0, -2338280408, -44506649036688]\) | \(-187778242790732059201/4984939585440150\) | \(-37534282066396685270400000000\) | \([2]\) | \(56623104\) | \(4.2653\) | |
| 58800.ei4 | 58800fz4 | \([0, -1, 0, -295372408, 1953463795312]\) | \(378499465220294881/120530818800\) | \(907541139264076800000000\) | \([2]\) | \(14155776\) | \(3.5721\) | |
| 58800.ei5 | 58800fz3 | \([0, -1, 0, -147980408, -677364236688]\) | \(47595748626367201/1215506250000\) | \(9152198067600000000000000\) | \([2, 2]\) | \(14155776\) | \(3.5721\) | |
| 58800.ei6 | 58800fz2 | \([0, -1, 0, -20972408, 21687795312]\) | \(135487869158881/51438240000\) | \(387306079856640000000000\) | \([2, 2]\) | \(7077888\) | \(3.2255\) | |
| 58800.ei7 | 58800fz1 | \([0, -1, 0, 4115592, 2420211312]\) | \(1023887723039/928972800\) | \(-6994734140620800000000\) | \([2]\) | \(3538944\) | \(2.8790\) | \(\Gamma_0(N)\)-optimal |
| 58800.ei8 | 58800fz5 | \([0, -1, 0, 24891592, -2165446412688]\) | \(226523624554079/269165039062500\) | \(-2026687851562500000000000000\) | \([2]\) | \(28311552\) | \(3.9187\) |
Rank
sage: E.rank()
The elliptic curves in class 58800.ei have rank \(1\).
Complex multiplication
The elliptic curves in class 58800.ei do not have complex multiplication.Modular form 58800.2.a.ei
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.
