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SageMath
E = EllipticCurve("q1")
E.isogeny_class()
Elliptic curves in class 4680.q
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
4680.q1 | 4680r3 | \([0, 0, 0, -89067, 4870726]\) | \(52337949619538/23423590125\) | \(34971232667904000\) | \([2]\) | \(24576\) | \(1.8699\) | |
4680.q2 | 4680r2 | \([0, 0, 0, -44067, -3508274]\) | \(12677589459076/213890625\) | \(159668496000000\) | \([2, 2]\) | \(12288\) | \(1.5233\) | |
4680.q3 | 4680r1 | \([0, 0, 0, -43887, -3538766]\) | \(50091484483024/14625\) | \(2729376000\) | \([2]\) | \(6144\) | \(1.1767\) | \(\Gamma_0(N)\)-optimal |
4680.q4 | 4680r4 | \([0, 0, 0, -1947, -9935786]\) | \(-546718898/28564453125\) | \(-42646500000000000\) | \([2]\) | \(24576\) | \(1.8699\) |
Rank
sage: E.rank()
The elliptic curves in class 4680.q have rank \(0\).
Complex multiplication
The elliptic curves in class 4680.q do not have complex multiplication.Modular form 4680.2.a.q
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.