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SageMath
sage: E = EllipticCurve("y1")
sage: E.isogeny_class()
Elliptic curves in class 338130y
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | Torsion structure | Modular degree | Optimality |
---|---|---|---|---|---|
338130.y7 | 338130y1 | [1, -1, 0, -5452075800, -149202592145600] | [2] | 637009920 | \(\Gamma_0(N)\)-optimal |
338130.y6 | 338130y2 | [1, -1, 0, -14454448920, 470482963468096] | [2, 2] | 1274019840 | |
338130.y5 | 338130y3 | [1, -1, 0, -67087036440, 6644606911560256] | [2] | 1911029760 | |
338130.y4 | 338130y4 | [1, -1, 0, -211651864920, 37473670967754496] | [2] | 2548039680 | |
338130.y8 | 338130y5 | [1, -1, 0, 38704997160, 3127062489980800] | [2] | 2548039680 | |
338130.y2 | 338130y6 | [1, -1, 0, -1071449453160, 426880086563858800] | [2, 2] | 3822059520 | |
338130.y1 | 338130y7 | [1, -1, 0, -17143191015660, 27320313468836296300] | [2] | 7644119040 | |
338130.y3 | 338130y8 | [1, -1, 0, -1069506558180, 428505349689527476] | [2] | 7644119040 |
Rank
sage: E.rank()
The elliptic curves in class 338130y have rank \(1\).
Complex multiplication
The elliptic curves in class 338130y do not have complex multiplication.Modular form 338130.2.a.y
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 Cremona numbering.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.