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SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 3038.j
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
3038.j1 | 3038i3 | \([1, -1, 1, -16204, -789855]\) | \(3999236143617/62\) | \(7294238\) | \([2]\) | \(3072\) | \(0.86488\) | |
3038.j2 | 3038i4 | \([1, -1, 1, -1504, 1201]\) | \(3196010817/1847042\) | \(217302644258\) | \([2]\) | \(3072\) | \(0.86488\) | |
3038.j3 | 3038i2 | \([1, -1, 1, -1014, -12127]\) | \(979146657/3844\) | \(452242756\) | \([2, 2]\) | \(1536\) | \(0.51830\) | |
3038.j4 | 3038i1 | \([1, -1, 1, -34, -367]\) | \(-35937/496\) | \(-58353904\) | \([2]\) | \(768\) | \(0.17173\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 3038.j have rank \(0\).
Complex multiplication
The elliptic curves in class 3038.j do not have complex multiplication.Modular form 3038.2.a.j
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.