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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 85697.a
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 85697.a1 | 85697a4 | \([1, -1, 1, -457156, 119086142]\) | \(82483294977/17\) | \(2177704826657\) | \([2]\) | \(361760\) | \(1.7547\) | |
| 85697.a2 | 85697a2 | \([1, -1, 1, -28671, 1852646]\) | \(20346417/289\) | \(37020982053169\) | \([2, 2]\) | \(180880\) | \(1.4081\) | |
| 85697.a3 | 85697a1 | \([1, -1, 1, -3466, -32688]\) | \(35937/17\) | \(2177704826657\) | \([2]\) | \(90440\) | \(1.0616\) | \(\Gamma_0(N)\)-optimal |
| 85697.a4 | 85697a3 | \([1, -1, 1, -3466, 4978066]\) | \(-35937/83521\) | \(-10699063813365841\) | \([2]\) | \(361760\) | \(1.7547\) |
Rank
sage: E.rank()
The elliptic curves in class 85697.a have rank \(0\).
Complex multiplication
The elliptic curves in class 85697.a do not have complex multiplication.Modular form 85697.2.a.a
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.
