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SageMath
E = EllipticCurve("y1")
E.isogeny_class()
Elliptic curves in class 3136y
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 3136.z5 | 3136y1 | \([0, -1, 0, -1633, 51969]\) | \(-15625/28\) | \(-863547424768\) | \([2]\) | \(3072\) | \(0.98059\) | \(\Gamma_0(N)\)-optimal |
| 3136.z4 | 3136y2 | \([0, -1, 0, -32993, 2316161]\) | \(128787625/98\) | \(3022415986688\) | \([2]\) | \(6144\) | \(1.3272\) | |
| 3136.z6 | 3136y3 | \([0, -1, 0, 14047, -1080127]\) | \(9938375/21952\) | \(-677021181018112\) | \([2]\) | \(9216\) | \(1.5299\) | |
| 3136.z3 | 3136y4 | \([0, -1, 0, -111393, -11692351]\) | \(4956477625/941192\) | \(29027283136151552\) | \([2]\) | \(18432\) | \(1.8765\) | |
| 3136.z2 | 3136y5 | \([0, -1, 0, -534753, -150770815]\) | \(-548347731625/1835008\) | \(-56593444029595648\) | \([2]\) | \(27648\) | \(2.0792\) | |
| 3136.z1 | 3136y6 | \([0, -1, 0, -8562913, -9641661567]\) | \(2251439055699625/25088\) | \(773738492592128\) | \([2]\) | \(55296\) | \(2.4258\) |
Rank
sage: E.rank()
The elliptic curves in class 3136y have rank \(1\).
Complex multiplication
The elliptic curves in class 3136y do not have complex multiplication.Modular form 3136.2.a.y
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 3 & 6 & 9 & 18 \\ 2 & 1 & 6 & 3 & 18 & 9 \\ 3 & 6 & 1 & 2 & 3 & 6 \\ 6 & 3 & 2 & 1 & 6 & 3 \\ 9 & 18 & 3 & 6 & 1 & 2 \\ 18 & 9 & 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
