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SageMath
E = EllipticCurve("ba1")
E.isogeny_class()
Elliptic curves in class 6762ba
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
6762.z2 | 6762ba1 | \([1, 1, 1, -20973, -1189917]\) | \(-25282750375/304704\) | \(-12295905467328\) | \([2]\) | \(16128\) | \(1.3234\) | \(\Gamma_0(N)\)-optimal |
6762.z1 | 6762ba2 | \([1, 1, 1, -336533, -75283405]\) | \(104453838382375/14904\) | \(601430158728\) | \([2]\) | \(32256\) | \(1.6699\) |
Rank
sage: E.rank()
The elliptic curves in class 6762ba have rank \(0\).
Complex multiplication
The elliptic curves in class 6762ba do not have complex multiplication.Modular form 6762.2.a.ba
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.