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SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 285.c
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
285.c1 | 285c3 | \([1, 1, 0, -6007, -181724]\) | \(23977812996389881/146611125\) | \(146611125\) | \([2]\) | \(288\) | \(0.75414\) | |
285.c2 | 285c4 | \([1, 1, 0, -1237, 13054]\) | \(209595169258201/41748046875\) | \(41748046875\) | \([4]\) | \(288\) | \(0.75414\) | |
285.c3 | 285c2 | \([1, 1, 0, -382, -2849]\) | \(6189976379881/456890625\) | \(456890625\) | \([2, 2]\) | \(144\) | \(0.40757\) | |
285.c4 | 285c1 | \([1, 1, 0, 23, -176]\) | \(1256216039/15582375\) | \(-15582375\) | \([2]\) | \(72\) | \(0.060992\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 285.c have rank \(0\).
Complex multiplication
The elliptic curves in class 285.c do not have complex multiplication.Modular form 285.2.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.