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SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 8450.i
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
8450.i1 | 8450g3 | \([1, 1, 0, -876775, -303676875]\) | \(988345570681/44994560\) | \(3393439799360000000\) | \([2]\) | \(290304\) | \(2.3176\) | |
8450.i2 | 8450g1 | \([1, 1, 0, -137400, 19430000]\) | \(3803721481/26000\) | \(1960891156250000\) | \([2]\) | \(96768\) | \(1.7683\) | \(\Gamma_0(N)\)-optimal |
8450.i3 | 8450g2 | \([1, 1, 0, -52900, 43174500]\) | \(-217081801/10562500\) | \(-796612032226562500\) | \([2]\) | \(193536\) | \(2.1149\) | |
8450.i4 | 8450g4 | \([1, 1, 0, 475225, -1154084875]\) | \(157376536199/7722894400\) | \(-582452128062025000000\) | \([2]\) | \(580608\) | \(2.6642\) |
Rank
sage: E.rank()
The elliptic curves in class 8450.i have rank \(1\).
Complex multiplication
The elliptic curves in class 8450.i do not have complex multiplication.Modular form 8450.2.a.i
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.