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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 253704e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
253704.e4 | 253704e1 | \([0, -1, 0, 641, -171236]\) | \(2048/891\) | \(-12652252476336\) | \([2]\) | \(737280\) | \(1.1932\) | \(\Gamma_0(N)\)-optimal |
253704.e3 | 253704e2 | \([0, -1, 0, -42604, -3284876]\) | \(37642192/1089\) | \(247421826203904\) | \([2, 2]\) | \(1474560\) | \(1.5397\) | |
253704.e2 | 253704e3 | \([0, -1, 0, -100264, 7578268]\) | \(122657188/43923\) | \(39917387960896512\) | \([2]\) | \(2949120\) | \(1.8863\) | |
253704.e1 | 253704e4 | \([0, -1, 0, -676864, -214112900]\) | \(37736227588/33\) | \(29990524388352\) | \([2]\) | \(2949120\) | \(1.8863\) |
Rank
sage: E.rank()
The elliptic curves in class 253704e have rank \(1\).
Complex multiplication
The elliptic curves in class 253704e do not have complex multiplication.Modular form 253704.2.a.e
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.