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SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 370881r
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality | CM discriminant |
---|---|---|---|---|---|---|---|---|---|
370881.r1 | 370881r1 | \([0, 0, 1, 0, -1237742]\) | \(0\) | \(-661826004347403\) | \([]\) | \(1064880\) | \(1.5227\) | \(\Gamma_0(N)\)-optimal | \(-3\) |
370881.r2 | 370881r2 | \([0, 0, 1, 0, 33419027]\) | \(0\) | \(-482471157169256787\) | \([]\) | \(3194640\) | \(2.0720\) | \(-3\) |
Rank
sage: E.rank()
The elliptic curves in class 370881r have rank \(1\).
Complex multiplication
Each elliptic curve in class 370881r has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-3}) \).Modular form 370881.2.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.