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SageMath
E = EllipticCurve("q1")
E.isogeny_class()
Elliptic curves in class 5775.q
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 5775.q1 | 5775c5 | \([1, 1, 0, -76230000, -256206911625]\) | \(3135316978843283198764801/571725\) | \(8933203125\) | \([2]\) | \(184320\) | \(2.7039\) | |
| 5775.q2 | 5775c3 | \([1, 1, 0, -4764375, -4004721000]\) | \(765458482133960722801/326869475625\) | \(5107335556640625\) | \([2, 2]\) | \(92160\) | \(2.3573\) | |
| 5775.q3 | 5775c6 | \([1, 1, 0, -4740750, -4046371875]\) | \(-754127868744065783521/15825714261328125\) | \(-247276785333251953125\) | \([2]\) | \(184320\) | \(2.7039\) | |
| 5775.q4 | 5775c4 | \([1, 1, 0, -636125, 102716250]\) | \(1821931919215868881/761147600816295\) | \(11892931262754609375\) | \([2]\) | \(92160\) | \(2.3573\) | |
| 5775.q5 | 5775c2 | \([1, 1, 0, -299250, -62015625]\) | \(189674274234120481/3859869269025\) | \(60310457328515625\) | \([2, 2]\) | \(46080\) | \(2.0107\) | |
| 5775.q6 | 5775c1 | \([1, 1, 0, 875, -2891000]\) | \(4733169839/231139696095\) | \(-3611557751484375\) | \([2]\) | \(23040\) | \(1.6641\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 5775.q have rank \(1\).
Complex multiplication
The elliptic curves in class 5775.q do not have complex multiplication.Modular form 5775.2.a.q
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.
