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SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 1710.i
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
1710.i1 | 1710h3 | \([1, -1, 0, -5589, 161995]\) | \(26487576322129/44531250\) | \(32463281250\) | \([2]\) | \(2048\) | \(0.91222\) | |
1710.i2 | 1710h2 | \([1, -1, 0, -459, 913]\) | \(14688124849/8122500\) | \(5921302500\) | \([2, 2]\) | \(1024\) | \(0.56565\) | |
1710.i3 | 1710h1 | \([1, -1, 0, -279, -1715]\) | \(3301293169/22800\) | \(16621200\) | \([2]\) | \(512\) | \(0.21908\) | \(\Gamma_0(N)\)-optimal |
1710.i4 | 1710h4 | \([1, -1, 0, 1791, 5863]\) | \(871257511151/527800050\) | \(-384766236450\) | \([2]\) | \(2048\) | \(0.91222\) |
Rank
sage: E.rank()
The elliptic curves in class 1710.i have rank \(1\).
Complex multiplication
The elliptic curves in class 1710.i do not have complex multiplication.Modular form 1710.2.a.i
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.