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SageMath
E = EllipticCurve("jr1")
E.isogeny_class()
Elliptic curves in class 221760.jr
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 221760.jr1 | 221760cd6 | \([0, 0, 0, -60655692, -181820644976]\) | \(258286045443018193442/8440380939375\) | \(806490958044119040000\) | \([2]\) | \(16777216\) | \(3.1055\) | |
| 221760.jr2 | 221760cd3 | \([0, 0, 0, -17136012, 27302889616]\) | \(11647843478225136004/128410942275\) | \(6134928304929177600\) | \([2]\) | \(8388608\) | \(2.7590\) | |
| 221760.jr3 | 221760cd4 | \([0, 0, 0, -3955692, -2580604976]\) | \(143279368983686884/22699269140625\) | \(1084474471449600000000\) | \([2, 2]\) | \(8388608\) | \(2.7590\) | |
| 221760.jr4 | 221760cd2 | \([0, 0, 0, -1098012, 403956016]\) | \(12257375872392016/1191317675625\) | \(14229022073333760000\) | \([2, 2]\) | \(4194304\) | \(2.4124\) | |
| 221760.jr5 | 221760cd1 | \([0, 0, 0, 82968, 30293944]\) | \(84611246065664/580054565475\) | \(-433008412908825600\) | \([2]\) | \(2097152\) | \(2.0658\) | \(\Gamma_0(N)\)-optimal |
| 221760.jr6 | 221760cd5 | \([0, 0, 0, 7021428, -14352468464]\) | \(400647648358480318/1163177490234375\) | \(-111143340000000000000000\) | \([2]\) | \(16777216\) | \(3.1055\) |
Rank
sage: E.rank()
The elliptic curves in class 221760.jr have rank \(1\).
Complex multiplication
The elliptic curves in class 221760.jr do not have complex multiplication.Modular form 221760.2.a.jr
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 2 & 4 & 8 & 4 \\ 8 & 1 & 4 & 2 & 4 & 8 \\ 2 & 4 & 1 & 2 & 4 & 2 \\ 4 & 2 & 2 & 1 & 2 & 4 \\ 8 & 4 & 4 & 2 & 1 & 8 \\ 4 & 8 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
