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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 130.a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
130.a1 | 130a3 | \([1, 0, 1, -208, -1122]\) | \(988345570681/44994560\) | \(44994560\) | \([2]\) | \(72\) | \(0.23045\) | |
130.a2 | 130a1 | \([1, 0, 1, -33, 68]\) | \(3803721481/26000\) | \(26000\) | \([6]\) | \(24\) | \(-0.31886\) | \(\Gamma_0(N)\)-optimal |
130.a3 | 130a2 | \([1, 0, 1, -13, 156]\) | \(-217081801/10562500\) | \(-10562500\) | \([6]\) | \(48\) | \(0.027717\) | |
130.a4 | 130a4 | \([1, 0, 1, 112, -4194]\) | \(157376536199/7722894400\) | \(-7722894400\) | \([2]\) | \(144\) | \(0.57702\) |
Rank
sage: E.rank()
The elliptic curves in class 130.a have rank \(1\).
Complex multiplication
The elliptic curves in class 130.a do not have complex multiplication.Modular form 130.2.a.a
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.