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SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 8470.r
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 8470.r1 | 8470u4 | \([1, 0, 0, -1893471, -774028199]\) | \(423783056881319689/99207416000000\) | \(175751989096376000000\) | \([2]\) | \(414720\) | \(2.5965\) | |
| 8470.r2 | 8470u2 | \([1, 0, 0, -1770656, -907026680]\) | \(346553430870203929/8300600\) | \(14705019236600\) | \([2]\) | \(138240\) | \(2.0472\) | |
| 8470.r3 | 8470u1 | \([1, 0, 0, -110536, -14214144]\) | \(-84309998289049/414124480\) | \(-733646777913280\) | \([2]\) | \(69120\) | \(1.7006\) | \(\Gamma_0(N)\)-optimal |
| 8470.r4 | 8470u3 | \([1, 0, 0, 274849, -75395495]\) | \(1296134247276791/2137096192000\) | \(-3785996266995712000\) | \([2]\) | \(207360\) | \(2.2499\) |
Rank
sage: E.rank()
The elliptic curves in class 8470.r have rank \(1\).
Complex multiplication
The elliptic curves in class 8470.r 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 LMFDB numbering.
\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
