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SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 8470r
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
8470.bb1 | 8470r1 | \([1, 0, 0, -41, -175]\) | \(-63088729/68600\) | \(-8300600\) | \([]\) | \(1728\) | \(0.022461\) | \(\Gamma_0(N)\)-optimal |
8470.bb2 | 8470r2 | \([1, 0, 0, 344, 3136]\) | \(37199299511/56000000\) | \(-6776000000\) | \([]\) | \(5184\) | \(0.57177\) |
Rank
sage: E.rank()
The elliptic curves in class 8470r have rank \(1\).
Complex multiplication
The elliptic curves in class 8470r do not have complex multiplication.Modular form 8470.2.a.r
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.