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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 990.b
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 990.b1 | 990c5 | \([1, -1, 0, -1539765, 735795481]\) | \(553808571467029327441/12529687500\) | \(9134142187500\) | \([2]\) | \(12288\) | \(2.0111\) | |
| 990.b2 | 990c3 | \([1, -1, 0, -106425, -13307459]\) | \(182864522286982801/463015182960\) | \(337538068377840\) | \([2]\) | \(6144\) | \(1.6645\) | |
| 990.b3 | 990c4 | \([1, -1, 0, -96345, 11487325]\) | \(135670761487282321/643043610000\) | \(468778791690000\) | \([2, 2]\) | \(6144\) | \(1.6645\) | |
| 990.b4 | 990c6 | \([1, -1, 0, -46845, 23238625]\) | \(-15595206456730321/310672490129100\) | \(-226480245304113900\) | \([2]\) | \(12288\) | \(2.0111\) | |
| 990.b5 | 990c2 | \([1, -1, 0, -9225, -29939]\) | \(119102750067601/68309049600\) | \(49797297158400\) | \([2, 2]\) | \(3072\) | \(1.3179\) | |
| 990.b6 | 990c1 | \([1, -1, 0, 2295, -4595]\) | \(1833318007919/1070530560\) | \(-780416778240\) | \([2]\) | \(1536\) | \(0.97134\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 990.b have rank \(0\).
Complex multiplication
The elliptic curves in class 990.b do not have complex multiplication.Modular form 990.2.a.b
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 2 & 4 & 4 & 8 \\ 8 & 1 & 4 & 8 & 2 & 4 \\ 2 & 4 & 1 & 2 & 2 & 4 \\ 4 & 8 & 2 & 1 & 4 & 8 \\ 4 & 2 & 2 & 4 & 1 & 2 \\ 8 & 4 & 4 & 8 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
