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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 400.e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
400.e1 | 400a3 | \([0, 0, 0, -2675, 53250]\) | \(132304644/5\) | \(80000000\) | \([4]\) | \(192\) | \(0.60250\) | |
400.e2 | 400a2 | \([0, 0, 0, -175, 750]\) | \(148176/25\) | \(100000000\) | \([2, 2]\) | \(96\) | \(0.25593\) | |
400.e3 | 400a1 | \([0, 0, 0, -50, -125]\) | \(55296/5\) | \(1250000\) | \([2]\) | \(48\) | \(-0.090642\) | \(\Gamma_0(N)\)-optimal |
400.e4 | 400a4 | \([0, 0, 0, 325, 4250]\) | \(237276/625\) | \(-10000000000\) | \([2]\) | \(192\) | \(0.60250\) |
Rank
sage: E.rank()
The elliptic curves in class 400.e have rank \(1\).
Complex multiplication
The elliptic curves in class 400.e do not have complex multiplication.Modular form 400.2.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.