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SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 26775.j
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 26775.j1 | 26775bg4 | \([1, -1, 1, -228380, -32360128]\) | \(115650783909361/27072079335\) | \(308367903675234375\) | \([2]\) | \(294912\) | \(2.0677\) | |
| 26775.j2 | 26775bg2 | \([1, -1, 1, -76505, 7734872]\) | \(4347507044161/258084225\) | \(2939740625390625\) | \([2, 2]\) | \(147456\) | \(1.7211\) | |
| 26775.j3 | 26775bg1 | \([1, -1, 1, -75380, 7984622]\) | \(4158523459441/16065\) | \(182990390625\) | \([4]\) | \(73728\) | \(1.3745\) | \(\Gamma_0(N)\)-optimal |
| 26775.j4 | 26775bg3 | \([1, -1, 1, 57370, 31832372]\) | \(1833318007919/39525924375\) | \(-450224982333984375\) | \([2]\) | \(294912\) | \(2.0677\) |
Rank
sage: E.rank()
The elliptic curves in class 26775.j have rank \(1\).
Complex multiplication
The elliptic curves in class 26775.j do not have complex multiplication.Modular form 26775.2.a.j
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
