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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 5712.e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
5712.e1 | 5712b3 | \([0, -1, 0, -544, 4624]\) | \(17418812548/1753941\) | \(1796035584\) | \([4]\) | \(2560\) | \(0.51187\) | |
5712.e2 | 5712b2 | \([0, -1, 0, -124, -416]\) | \(830321872/127449\) | \(32626944\) | \([2, 2]\) | \(1280\) | \(0.16530\) | |
5712.e3 | 5712b1 | \([0, -1, 0, -119, -462]\) | \(11745974272/357\) | \(5712\) | \([2]\) | \(640\) | \(-0.18127\) | \(\Gamma_0(N)\)-optimal |
5712.e4 | 5712b4 | \([0, -1, 0, 216, -2592]\) | \(1083360092/3306177\) | \(-3385525248\) | \([2]\) | \(2560\) | \(0.51187\) |
Rank
sage: E.rank()
The elliptic curves in class 5712.e have rank \(0\).
Complex multiplication
The elliptic curves in class 5712.e do not have complex multiplication.Modular form 5712.2.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 LMFDB 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 LMFDB labels.