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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 222.e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
222.e1 | 222a2 | \([1, 0, 0, -148, -706]\) | \(-358667682625/303918\) | \(-303918\) | \([]\) | \(36\) | \(-0.020627\) | |
222.e2 | 222a1 | \([1, 0, 0, 2, -4]\) | \(857375/7992\) | \(-7992\) | \([3]\) | \(12\) | \(-0.56993\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 222.e have rank \(0\).
Complex multiplication
The elliptic curves in class 222.e do not have complex multiplication.Modular form 222.2.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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.