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SageMath
E = EllipticCurve("cf1")
E.isogeny_class()
Elliptic curves in class 121275.cf
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 121275.cf1 | 121275eq6 | \([1, -1, 1, -49822205, -135345043578]\) | \(10206027697760497/5557167\) | \(7447135270925109375\) | \([2]\) | \(7864320\) | \(2.9498\) | |
| 121275.cf2 | 121275eq4 | \([1, -1, 1, -3131330, -2089286328]\) | \(2533811507137/58110129\) | \(77873130549056390625\) | \([2, 2]\) | \(3932160\) | \(2.6032\) | |
| 121275.cf3 | 121275eq2 | \([1, -1, 1, -430205, 60809172]\) | \(6570725617/2614689\) | \(3503933330490140625\) | \([2, 2]\) | \(1966080\) | \(2.2566\) | |
| 121275.cf4 | 121275eq1 | \([1, -1, 1, -375080, 88481922]\) | \(4354703137/1617\) | \(2166934650890625\) | \([2]\) | \(983040\) | \(1.9100\) | \(\Gamma_0(N)\)-optimal |
| 121275.cf5 | 121275eq5 | \([1, -1, 1, 341545, -6472054578]\) | \(3288008303/13504609503\) | \(-18097468323313247859375\) | \([2]\) | \(7864320\) | \(2.9498\) | |
| 121275.cf6 | 121275eq3 | \([1, -1, 1, 1388920, 439187172]\) | \(221115865823/190238433\) | \(-254937694742631140625\) | \([2]\) | \(3932160\) | \(2.6032\) |
Rank
sage: E.rank()
The elliptic curves in class 121275.cf have rank \(0\).
Complex multiplication
The elliptic curves in class 121275.cf do not have complex multiplication.Modular form 121275.2.a.cf
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 4 & 8 \\ 2 & 1 & 2 & 4 & 2 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 8 & 4 & 2 & 1 & 8 & 4 \\ 4 & 2 & 4 & 8 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
