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SageMath
E = EllipticCurve("cr1")
E.isogeny_class()
Elliptic curves in class 112896cr
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
112896.x2 | 112896cr1 | \([0, 0, 0, -234906, -44068640]\) | \(-881422385472/5764801\) | \(-9375755759064576\) | \([2]\) | \(688128\) | \(1.9009\) | \(\Gamma_0(N)\)-optimal |
112896.x1 | 112896cr2 | \([0, 0, 0, -3764376, -2811173120]\) | \(56676204750528/2401\) | \(249916021899264\) | \([2]\) | \(1376256\) | \(2.2475\) |
Rank
sage: E.rank()
The elliptic curves in class 112896cr have rank \(1\).
Complex multiplication
The elliptic curves in class 112896cr do not have complex multiplication.Modular form 112896.2.a.cr
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 Cremona numbering.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.