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SageMath
E = EllipticCurve("ec1")
E.isogeny_class()
Elliptic curves in class 100800.ec
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 100800.ec1 | 100800cx8 | \([0, 0, 0, -92894700, -216809714000]\) | \(29689921233686449/10380965400750\) | \(30997396591193088000000000\) | \([2]\) | \(21233664\) | \(3.5928\) | |
| 100800.ec2 | 100800cx5 | \([0, 0, 0, -82958700, -290831186000]\) | \(21145699168383889/2593080\) | \(7742895390720000000\) | \([2]\) | \(7077888\) | \(3.0435\) | |
| 100800.ec3 | 100800cx6 | \([0, 0, 0, -38894700, 90882286000]\) | \(2179252305146449/66177562500\) | \(197605142784000000000000\) | \([2, 2]\) | \(10616832\) | \(3.2462\) | |
| 100800.ec4 | 100800cx3 | \([0, 0, 0, -38606700, 92329774000]\) | \(2131200347946769/2058000\) | \(6145155072000000000\) | \([2]\) | \(5308416\) | \(2.8996\) | |
| 100800.ec5 | 100800cx2 | \([0, 0, 0, -5198700, -4518866000]\) | \(5203798902289/57153600\) | \(170659735142400000000\) | \([2, 2]\) | \(3538944\) | \(2.6969\) | |
| 100800.ec6 | 100800cx4 | \([0, 0, 0, -1166700, -11349074000]\) | \(-58818484369/18600435000\) | \(-55540601303040000000000\) | \([2]\) | \(7077888\) | \(3.0435\) | |
| 100800.ec7 | 100800cx1 | \([0, 0, 0, -590700, 61486000]\) | \(7633736209/3870720\) | \(11557907988480000000\) | \([2]\) | \(1769472\) | \(2.3503\) | \(\Gamma_0(N)\)-optimal |
| 100800.ec8 | 100800cx7 | \([0, 0, 0, 10497300, 305935054000]\) | \(42841933504271/13565917968750\) | \(-40507614000000000000000000\) | \([2]\) | \(21233664\) | \(3.5928\) |
Rank
sage: E.rank()
The elliptic curves in class 100800.ec have rank \(0\).
Complex multiplication
The elliptic curves in class 100800.ec do not have complex multiplication.Modular form 100800.2.a.ec
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 3 & 2 & 4 & 6 & 12 & 12 & 4 \\ 3 & 1 & 6 & 12 & 2 & 4 & 4 & 12 \\ 2 & 6 & 1 & 2 & 3 & 6 & 6 & 2 \\ 4 & 12 & 2 & 1 & 6 & 12 & 3 & 4 \\ 6 & 2 & 3 & 6 & 1 & 2 & 2 & 6 \\ 12 & 4 & 6 & 12 & 2 & 1 & 4 & 3 \\ 12 & 4 & 6 & 3 & 2 & 4 & 1 & 12 \\ 4 & 12 & 2 & 4 & 6 & 3 & 12 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
