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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 55470.d
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 55470.d1 | 55470c3 | \([1, 1, 0, -63431832, 84443489676]\) | \(4465136636671380769/2096375976562500\) | \(13251953635053477539062500\) | \([2]\) | \(15966720\) | \(3.5148\) | |
| 55470.d2 | 55470c1 | \([1, 1, 0, -32479572, -71256562416]\) | \(599437478278595809/33854760000\) | \(214008228896763240000\) | \([2]\) | \(5322240\) | \(2.9655\) | \(\Gamma_0(N)\)-optimal |
| 55470.d3 | 55470c2 | \([1, 1, 0, -30630572, -79724612616]\) | \(-502780379797811809/143268096832200\) | \(-905649653415623033377800\) | \([2]\) | \(10644480\) | \(3.3121\) | |
| 55470.d4 | 55470c4 | \([1, 1, 0, 225474418, 639663520926]\) | \(200541749524551119231/144008551960031250\) | \(-910330339100138068635281250\) | \([2]\) | \(31933440\) | \(3.8614\) |
Rank
sage: E.rank()
The elliptic curves in class 55470.d have rank \(1\).
Complex multiplication
The elliptic curves in class 55470.d do not have complex multiplication.Modular form 55470.2.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.
\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
