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SageMath
E = EllipticCurve("eb1")
E.isogeny_class()
Elliptic curves in class 129360eb
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 129360.a6 | 129360eb1 | \([0, -1, 0, 199904, 3202816]\) | \(1833318007919/1070530560\) | \(-515878296999690240\) | \([2]\) | \(1769472\) | \(2.0881\) | \(\Gamma_0(N)\)-optimal |
| 129360.a5 | 129360eb2 | \([0, -1, 0, -803616, 26484480]\) | \(119102750067601/68309049600\) | \(32917468677695078400\) | \([2, 2]\) | \(3538944\) | \(2.4347\) | |
| 129360.a3 | 129360eb3 | \([0, -1, 0, -8392736, -9317240064]\) | \(135670761487282321/643043610000\) | \(309876480708157440000\) | \([2, 2]\) | \(7077888\) | \(2.7813\) | |
| 129360.a2 | 129360eb4 | \([0, -1, 0, -9270816, 10844179200]\) | \(182864522286982801/463015182960\) | \(223122527273210019840\) | \([2]\) | \(7077888\) | \(2.7813\) | |
| 129360.a4 | 129360eb5 | \([0, -1, 0, -4080736, -18882980864]\) | \(-15595206456730321/310672490129100\) | \(-149710060712748998246400\) | \([2]\) | \(14155776\) | \(3.1279\) | |
| 129360.a1 | 129360eb6 | \([0, -1, 0, -134130656, -597871296000]\) | \(553808571467029327441/12529687500\) | \(6037934918400000000\) | \([2]\) | \(14155776\) | \(3.1279\) |
Rank
sage: E.rank()
The elliptic curves in class 129360eb have rank \(1\).
Complex multiplication
The elliptic curves in class 129360eb do not have complex multiplication.Modular form 129360.2.a.eb
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
