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SageMath
E = EllipticCurve("q1")
E.isogeny_class()
Elliptic curves in class 12705.q
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
12705.q1 | 12705p2 | \([0, 1, 1, -3241630, 42469124539]\) | \(-2126464142970105856/438611057788643355\) | \(-777026244147106810627155\) | \([]\) | \(3600000\) | \(3.2634\) | |
12705.q2 | 12705p1 | \([0, 1, 1, -1081780, -507519941]\) | \(-79028701534867456/16987307596875\) | \(-30094051633627471875\) | \([]\) | \(720000\) | \(2.4587\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 12705.q have rank \(0\).
Complex multiplication
The elliptic curves in class 12705.q do not have complex multiplication.Modular form 12705.2.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.
\(\left(\begin{array}{rr} 1 & 5 \\ 5 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.