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SageMath
E = EllipticCurve("dr1")
E.isogeny_class()
Elliptic curves in class 388080dr
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
388080.dr2 | 388080dr1 | \([0, 0, 0, 1721517, -414695918]\) | \(668944031/475200\) | \(-400815315922039603200\) | \([]\) | \(16257024\) | \(2.6418\) | \(\Gamma_0(N)\)-optimal |
388080.dr1 | 388080dr2 | \([0, 0, 0, -19023123, 38821716178]\) | \(-902612375329/249562500\) | \(-210497626851418368000000\) | \([]\) | \(48771072\) | \(3.1911\) |
Rank
sage: E.rank()
The elliptic curves in class 388080dr have rank \(0\).
Complex multiplication
The elliptic curves in class 388080dr do not have complex multiplication.Modular form 388080.2.a.dr
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.