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SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 722.c
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
722.c1 | 722a2 | \([1, 0, 1, -33581, -2375576]\) | \(-246579625/512\) | \(-8695584276992\) | \([]\) | \(2052\) | \(1.3686\) | |
722.c2 | 722a1 | \([1, 0, 1, 714, -16080]\) | \(2375/8\) | \(-135868504328\) | \([3]\) | \(684\) | \(0.81932\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 722.c have rank \(1\).
Complex multiplication
The elliptic curves in class 722.c do not have complex multiplication.Modular form 722.2.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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.