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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 450.d
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
450.d1 | 450g8 | \([1, -1, 0, -1200042, 506291116]\) | \(16778985534208729/81000\) | \(922640625000\) | \([2]\) | \(4608\) | \(1.9182\) | |
450.d2 | 450g7 | \([1, -1, 0, -102042, 1733116]\) | \(10316097499609/5859375000\) | \(66741943359375000\) | \([2]\) | \(4608\) | \(1.9182\) | |
450.d3 | 450g6 | \([1, -1, 0, -75042, 7916116]\) | \(4102915888729/9000000\) | \(102515625000000\) | \([2, 2]\) | \(2304\) | \(1.5716\) | |
450.d4 | 450g4 | \([1, -1, 0, -64917, -6350009]\) | \(2656166199049/33750\) | \(384433593750\) | \([2]\) | \(1536\) | \(1.3689\) | |
450.d5 | 450g5 | \([1, -1, 0, -15417, 638491]\) | \(35578826569/5314410\) | \(60534451406250\) | \([2]\) | \(1536\) | \(1.3689\) | |
450.d6 | 450g2 | \([1, -1, 0, -4167, -92759]\) | \(702595369/72900\) | \(830376562500\) | \([2, 2]\) | \(768\) | \(1.0223\) | |
450.d7 | 450g3 | \([1, -1, 0, -3042, 212116]\) | \(-273359449/1536000\) | \(-17496000000000\) | \([2]\) | \(1152\) | \(1.2250\) | |
450.d8 | 450g1 | \([1, -1, 0, 333, -7259]\) | \(357911/2160\) | \(-24603750000\) | \([2]\) | \(384\) | \(0.67574\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 450.d have rank \(0\).
Complex multiplication
The elliptic curves in class 450.d do not have complex multiplication.Modular form 450.2.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.
\(\left(\begin{array}{rrrrrrrr} 1 & 4 & 2 & 12 & 3 & 6 & 4 & 12 \\ 4 & 1 & 2 & 3 & 12 & 6 & 4 & 12 \\ 2 & 2 & 1 & 6 & 6 & 3 & 2 & 6 \\ 12 & 3 & 6 & 1 & 4 & 2 & 12 & 4 \\ 3 & 12 & 6 & 4 & 1 & 2 & 12 & 4 \\ 6 & 6 & 3 & 2 & 2 & 1 & 6 & 2 \\ 4 & 4 & 2 & 12 & 12 & 6 & 1 & 3 \\ 12 & 12 & 6 & 4 & 4 & 2 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.