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SageMath
E = EllipticCurve("m1")
E.isogeny_class()
Elliptic curves in class 1800.m
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 1800.m1 | 1800s5 | \([0, 0, 0, -86475, 9787750]\) | \(3065617154/9\) | \(209952000000\) | \([2]\) | \(4096\) | \(1.4018\) | |
| 1800.m2 | 1800s3 | \([0, 0, 0, -14475, -670250]\) | \(28756228/3\) | \(34992000000\) | \([2]\) | \(2048\) | \(1.0552\) | |
| 1800.m3 | 1800s4 | \([0, 0, 0, -5475, 148750]\) | \(1556068/81\) | \(944784000000\) | \([2, 2]\) | \(2048\) | \(1.0552\) | |
| 1800.m4 | 1800s2 | \([0, 0, 0, -975, -8750]\) | \(35152/9\) | \(26244000000\) | \([2, 2]\) | \(1024\) | \(0.70867\) | |
| 1800.m5 | 1800s1 | \([0, 0, 0, 150, -875]\) | \(2048/3\) | \(-546750000\) | \([2]\) | \(512\) | \(0.36210\) | \(\Gamma_0(N)\)-optimal |
| 1800.m6 | 1800s6 | \([0, 0, 0, 3525, 589750]\) | \(207646/6561\) | \(-153055008000000\) | \([2]\) | \(4096\) | \(1.4018\) |
Rank
sage: E.rank()
The elliptic curves in class 1800.m have rank \(1\).
Complex multiplication
The elliptic curves in class 1800.m do not have complex multiplication.Modular form 1800.2.a.m
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 2 & 4 & 8 & 4 \\ 8 & 1 & 4 & 2 & 4 & 8 \\ 2 & 4 & 1 & 2 & 4 & 2 \\ 4 & 2 & 2 & 1 & 2 & 4 \\ 8 & 4 & 4 & 2 & 1 & 8 \\ 4 & 8 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
