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SageMath
E = EllipticCurve("y1")
E.isogeny_class()
Elliptic curves in class 51870.y
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
51870.y1 | 51870bd8 | \([1, 0, 1, -768208004, -8195392298998]\) | \(50137213659805457275731367898809/4113897879000\) | \(4113897879000\) | \([2]\) | \(7962624\) | \(3.2723\) | |
51870.y2 | 51870bd6 | \([1, 0, 1, -48013004, -128055986998]\) | \(12240533203187013248735018809/3506282465049000000\) | \(3506282465049000000\) | \([2, 2]\) | \(3981312\) | \(2.9257\) | |
51870.y3 | 51870bd7 | \([1, 0, 1, -47818004, -129147674998]\) | \(-12091997009671629064982138809/207252595706436249879000\) | \(-207252595706436249879000\) | \([2]\) | \(7962624\) | \(3.2723\) | |
51870.y4 | 51870bd5 | \([1, 0, 1, -9485039, -11240205244]\) | \(94371532824107026279203049/40995077600666342790\) | \(40995077600666342790\) | \([6]\) | \(2654208\) | \(2.7230\) | |
51870.y5 | 51870bd3 | \([1, 0, 1, -3013004, -1983986998]\) | \(3024980849878413455018809/50557689000000000000\) | \(50557689000000000000\) | \([2]\) | \(1990656\) | \(2.5791\) | |
51870.y6 | 51870bd2 | \([1, 0, 1, -687089, -116077264]\) | \(35872512095393194378249/14944558319037792900\) | \(14944558319037792900\) | \([2, 6]\) | \(1327104\) | \(2.3764\) | |
51870.y7 | 51870bd1 | \([1, 0, 1, -322589, 69234536]\) | \(3712533999213317890249/76090919904090000\) | \(76090919904090000\) | \([6]\) | \(663552\) | \(2.0298\) | \(\Gamma_0(N)\)-optimal |
51870.y8 | 51870bd4 | \([1, 0, 1, 2278861, -850446484]\) | \(1308812680909424992398551/1070002284841633041990\) | \(-1070002284841633041990\) | \([6]\) | \(2654208\) | \(2.7230\) |
Rank
sage: E.rank()
The elliptic curves in class 51870.y have rank \(1\).
Complex multiplication
The elliptic curves in class 51870.y do not have complex multiplication.Modular form 51870.2.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 LMFDB numbering.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 3 & 4 & 6 & 12 & 12 \\ 2 & 1 & 2 & 6 & 2 & 3 & 6 & 6 \\ 4 & 2 & 1 & 12 & 4 & 6 & 12 & 3 \\ 3 & 6 & 12 & 1 & 12 & 2 & 4 & 4 \\ 4 & 2 & 4 & 12 & 1 & 6 & 3 & 12 \\ 6 & 3 & 6 & 2 & 6 & 1 & 2 & 2 \\ 12 & 6 & 12 & 4 & 3 & 2 & 1 & 4 \\ 12 & 6 & 3 & 4 & 12 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.