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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 6930.d
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 6930.d1 | 6930i3 | \([1, -1, 0, -802898370, -8756469168620]\) | \(78519570041710065450485106721/96428056919040\) | \(70296053493980160\) | \([2]\) | \(1474560\) | \(3.4067\) | |
| 6930.d2 | 6930i5 | \([1, -1, 0, -236147490, 1278495746356]\) | \(1997773216431678333214187041/187585177195046990066400\) | \(136749594175189255758405600\) | \([2]\) | \(2949120\) | \(3.7533\) | |
| 6930.d3 | 6930i4 | \([1, -1, 0, -52439490, -123820900844]\) | \(21876183941534093095979041/3572502915711058560000\) | \(2604354625553361690240000\) | \([2, 2]\) | \(1474560\) | \(3.4067\) | |
| 6930.d4 | 6930i2 | \([1, -1, 0, -50181570, -136808005100]\) | \(19170300594578891358373921/671785075055001600\) | \(489731319715096166400\) | \([2, 2]\) | \(737280\) | \(3.0602\) | |
| 6930.d5 | 6930i1 | \([1, -1, 0, -2995650, -2337570284]\) | \(-4078208988807294650401/880065599546327040\) | \(-641567822069272412160\) | \([2]\) | \(368640\) | \(2.7136\) | \(\Gamma_0(N)\)-optimal |
| 6930.d6 | 6930i6 | \([1, -1, 0, 95141790, -694989970700]\) | \(130650216943167617311657439/361816948816603087500000\) | \(-263764555687303650787500000\) | \([2]\) | \(2949120\) | \(3.7533\) |
Rank
sage: E.rank()
The elliptic curves in class 6930.d have rank \(0\).
Complex multiplication
The elliptic curves in class 6930.d do not have complex multiplication.Modular form 6930.2.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 LMFDB numbering.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 4 & 2 & 4 & 8 \\ 8 & 1 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 2 & 4 & 2 & 1 & 2 & 4 \\ 4 & 8 & 4 & 2 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
