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SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 1170.i
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
1170.i1 | 1170l3 | \([1, -1, 1, -1868, 30287]\) | \(988345570681/44994560\) | \(32801034240\) | \([6]\) | \(1728\) | \(0.77976\) | |
1170.i2 | 1170l1 | \([1, -1, 1, -293, -1843]\) | \(3803721481/26000\) | \(18954000\) | \([2]\) | \(576\) | \(0.23045\) | \(\Gamma_0(N)\)-optimal |
1170.i3 | 1170l2 | \([1, -1, 1, -113, -4219]\) | \(-217081801/10562500\) | \(-7700062500\) | \([2]\) | \(1152\) | \(0.57702\) | |
1170.i4 | 1170l4 | \([1, -1, 1, 1012, 113231]\) | \(157376536199/7722894400\) | \(-5629990017600\) | \([6]\) | \(3456\) | \(1.1263\) |
Rank
sage: E.rank()
The elliptic curves in class 1170.i have rank \(0\).
Complex multiplication
The elliptic curves in class 1170.i do not have complex multiplication.Modular form 1170.2.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.