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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 59643d
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality | CM discriminant |
---|---|---|---|---|---|---|---|---|---|
59643.f3 | 59643d1 | \([0, 0, 1, 0, -25956]\) | \(0\) | \(-291038813883\) | \([]\) | \(35328\) | \(0.87861\) | \(\Gamma_0(N)\)-optimal | \(-3\) |
59643.f2 | 59643d2 | \([0, 0, 1, -66270, -6566805]\) | \(-12288000\) | \(-2619349324947\) | \([]\) | \(105984\) | \(1.4279\) | \(-27\) | |
59643.f4 | 59643d3 | \([0, 0, 1, 0, 700805]\) | \(0\) | \(-212167295320707\) | \([]\) | \(105984\) | \(1.4279\) | \(-3\) | |
59643.f1 | 59643d4 | \([0, 0, 1, -596430, 177303728]\) | \(-12288000\) | \(-1909505657886363\) | \([]\) | \(317952\) | \(1.9772\) | \(-27\) |
Rank
sage: E.rank()
The elliptic curves in class 59643d have rank \(1\).
Complex multiplication
Each elliptic curve in class 59643d has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-3}) \).Modular form 59643.2.a.d
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.
\(\left(\begin{array}{rrrr} 1 & 3 & 3 & 9 \\ 3 & 1 & 9 & 27 \\ 3 & 9 & 1 & 3 \\ 9 & 27 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.