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SageMath
E = EllipticCurve("qf1")
E.isogeny_class()
Elliptic curves in class 486720.qf
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
486720.qf1 | 486720qf3 | \([0, 0, 0, -8423032332, 297543738313744]\) | \(71647584155243142409/10140000\) | \(9353314103863541760000\) | \([2]\) | \(330301440\) | \(4.0688\) | \(\Gamma_0(N)\)-optimal* |
486720.qf2 | 486720qf4 | \([0, 0, 0, -604362252, 3182721592336]\) | \(26465989780414729/10571870144160\) | \(9751678710412783278033469440\) | \([2]\) | \(330301440\) | \(4.0688\) | |
486720.qf3 | 486720qf2 | \([0, 0, 0, -526487052, 4648239406096]\) | \(17496824387403529/6580454400\) | \(6069926720843284060569600\) | \([2, 2]\) | \(165150720\) | \(3.7222\) | \(\Gamma_0(N)\)-optimal* |
486720.qf4 | 486720qf1 | \([0, 0, 0, -28085772, 94645951504]\) | \(-2656166199049/2658140160\) | \(-2451915172443204291133440\) | \([2]\) | \(82575360\) | \(3.3756\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 486720.qf have rank \(0\).
Complex multiplication
The elliptic curves in class 486720.qf do not have complex multiplication.Modular form 486720.2.a.qf
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.