Show commands:
SageMath
E = EllipticCurve("cp1")
E.isogeny_class()
Elliptic curves in class 40425.cp
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 40425.cp1 | 40425cg6 | \([1, 0, 1, -3735270026, 87867764877323]\) | \(3135316978843283198764801/571725\) | \(1050982414453125\) | \([2]\) | \(8847360\) | \(3.6768\) | |
| 40425.cp2 | 40425cg4 | \([1, 0, 1, -233454401, 1372918939823]\) | \(765458482133960722801/326869475625\) | \(600872920903212890625\) | \([2, 2]\) | \(4423680\) | \(3.3302\) | |
| 40425.cp3 | 40425cg5 | \([1, 0, 1, -232296776, 1387208662823]\) | \(-754127868744065783521/15825714261328125\) | \(-29091866517671759033203125\) | \([2]\) | \(8847360\) | \(3.6768\) | |
| 40425.cp4 | 40425cg3 | \([1, 0, 1, -31170151, -35325184177]\) | \(1821931919215868881/761147600816295\) | \(1399191470131817038359375\) | \([2]\) | \(4423680\) | \(3.3302\) | |
| 40425.cp5 | 40425cg2 | \([1, 0, 1, -14663276, 21227369573]\) | \(189674274234120481/3859869269025\) | \(7095464994242534765625\) | \([2, 2]\) | \(2211840\) | \(2.9837\) | |
| 40425.cp6 | 40425cg1 | \([1, 0, 1, 42849, 991741573]\) | \(4733169839/231139696095\) | \(-424896157904385234375\) | \([2]\) | \(1105920\) | \(2.6371\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 40425.cp have rank \(1\).
Complex multiplication
The elliptic curves in class 40425.cp do not have complex multiplication.Modular form 40425.2.a.cp
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 4 & 8 \\ 2 & 1 & 2 & 4 & 2 & 4 \\ 4 & 2 & 1 & 8 & 4 & 8 \\ 8 & 4 & 8 & 1 & 2 & 4 \\ 4 & 2 & 4 & 2 & 1 & 2 \\ 8 & 4 & 8 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
