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SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 1785.c
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
1785.c1 | 1785k4 | \([1, 0, 0, -1686, 26505]\) | \(530044731605089/26309115\) | \(26309115\) | \([2]\) | \(1024\) | \(0.49515\) | |
1785.c2 | 1785k3 | \([1, 0, 0, -536, -4485]\) | \(17032120495489/1339001685\) | \(1339001685\) | \([2]\) | \(1024\) | \(0.49515\) | |
1785.c3 | 1785k2 | \([1, 0, 0, -111, 360]\) | \(151334226289/28676025\) | \(28676025\) | \([2, 2]\) | \(512\) | \(0.14858\) | |
1785.c4 | 1785k1 | \([1, 0, 0, 14, 35]\) | \(302111711/669375\) | \(-669375\) | \([2]\) | \(256\) | \(-0.19800\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 1785.c have rank \(1\).
Complex multiplication
The elliptic curves in class 1785.c do not have complex multiplication.Modular form 1785.2.a.c
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.