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SageMath
E = EllipticCurve("bb1")
E.isogeny_class()
Elliptic curves in class 10192bb
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
10192.g2 | 10192bb1 | \([0, 1, 0, -5749, 166179]\) | \(-43614208/91\) | \(-43852017664\) | \([]\) | \(13824\) | \(0.92759\) | \(\Gamma_0(N)\)-optimal |
10192.g3 | 10192bb2 | \([0, 1, 0, 9931, 837283]\) | \(224755712/753571\) | \(-363138558275584\) | \([]\) | \(41472\) | \(1.4769\) | |
10192.g1 | 10192bb3 | \([0, 1, 0, -91989, -26497661]\) | \(-178643795968/524596891\) | \(-252798155281444864\) | \([]\) | \(124416\) | \(2.0262\) |
Rank
sage: E.rank()
The elliptic curves in class 10192bb have rank \(1\).
Complex multiplication
The elliptic curves in class 10192bb do not have complex multiplication.Modular form 10192.2.a.bb
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.
\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.