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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 234e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
234.e3 | 234e1 | \([1, -1, 1, 4, -7]\) | \(12167/26\) | \(-18954\) | \([]\) | \(20\) | \(-0.49492\) | \(\Gamma_0(N)\)-optimal |
234.e2 | 234e2 | \([1, -1, 1, -41, 209]\) | \(-10218313/17576\) | \(-12812904\) | \([3]\) | \(60\) | \(0.054386\) | |
234.e1 | 234e3 | \([1, -1, 1, -4136, 103403]\) | \(-10730978619193/6656\) | \(-4852224\) | \([3]\) | \(180\) | \(0.60369\) |
Rank
sage: E.rank()
The elliptic curves in class 234e have rank \(0\).
Complex multiplication
The elliptic curves in class 234e do not have complex multiplication.Modular form 234.2.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.
\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.