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SageMath
E = EllipticCurve("q1")
E.isogeny_class()
Elliptic curves in class 1680.q
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
1680.q1 | 1680t5 | \([0, 1, 0, -268800, 53550900]\) | \(524388516989299201/3150\) | \(12902400\) | \([4]\) | \(6144\) | \(1.4299\) | |
1680.q2 | 1680t4 | \([0, 1, 0, -16800, 832500]\) | \(128031684631201/9922500\) | \(40642560000\) | \([2, 4]\) | \(3072\) | \(1.0833\) | |
1680.q3 | 1680t6 | \([0, 1, 0, -15680, 949428]\) | \(-104094944089921/35880468750\) | \(-146966400000000\) | \([8]\) | \(6144\) | \(1.4299\) | |
1680.q4 | 1680t3 | \([0, 1, 0, -5920, -167692]\) | \(5602762882081/345888060\) | \(1416757493760\) | \([2]\) | \(3072\) | \(1.0833\) | |
1680.q5 | 1680t2 | \([0, 1, 0, -1120, 10868]\) | \(37966934881/8643600\) | \(35404185600\) | \([2, 2]\) | \(1536\) | \(0.73676\) | |
1680.q6 | 1680t1 | \([0, 1, 0, 160, 1140]\) | \(109902239/188160\) | \(-770703360\) | \([2]\) | \(768\) | \(0.39018\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 1680.q have rank \(0\).
Complex multiplication
The elliptic curves in class 1680.q do not have complex multiplication.Modular form 1680.2.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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 4 & 8 \\ 2 & 1 & 2 & 4 & 2 & 4 \\ 4 & 2 & 1 & 8 & 4 & 8 \\ 8 & 4 & 8 & 1 & 2 & 4 \\ 4 & 2 & 4 & 2 & 1 & 2 \\ 8 & 4 & 8 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.