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SageMath
E = EllipticCurve("cr1")
E.isogeny_class()
Elliptic curves in class 69696cr
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 69696.bn3 | 69696cr1 | \([0, 0, 0, -144111, 21056420]\) | \(4004529472/99\) | \(8182741051584\) | \([2]\) | \(245760\) | \(1.5869\) | \(\Gamma_0(N)\)-optimal |
| 69696.bn2 | 69696cr2 | \([0, 0, 0, -149556, 19379360]\) | \(69934528/9801\) | \(51845847302836224\) | \([2, 2]\) | \(491520\) | \(1.9335\) | |
| 69696.bn4 | 69696cr3 | \([0, 0, 0, 242484, 103903184]\) | \(37259704/131769\) | \(-5576308909905051648\) | \([2]\) | \(983040\) | \(2.2800\) | |
| 69696.bn1 | 69696cr4 | \([0, 0, 0, -628716, -172476304]\) | \(649461896/72171\) | \(3054191732021624832\) | \([2]\) | \(983040\) | \(2.2800\) |
Rank
sage: E.rank()
The elliptic curves in class 69696cr have rank \(1\).
Complex multiplication
The elliptic curves in class 69696cr do not have complex multiplication.Modular form 69696.2.a.cr
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 Cremona 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 Cremona labels.
