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SageMath
E = EllipticCurve("bc1")
E.isogeny_class()
Elliptic curves in class 454860bc
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
454860.bc1 | 454860bc1 | \([0, 0, 0, -1836768, 958113133]\) | \(1248870793216/42525\) | \(23335302708219600\) | \([2]\) | \(6842880\) | \(2.2324\) | \(\Gamma_0(N)\)-optimal |
454860.bc2 | 454860bc2 | \([0, 0, 0, -1755543, 1046697118]\) | \(-68150496976/14467005\) | \(-127018719701380926720\) | \([2]\) | \(13685760\) | \(2.5790\) |
Rank
sage: E.rank()
The elliptic curves in class 454860bc have rank \(1\).
Complex multiplication
The elliptic curves in class 454860bc do not have complex multiplication.Modular form 454860.2.a.bc
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.