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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 1040e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
1040.b1 | 1040e1 | \([0, 1, 0, -13456, 596244]\) | \(65787589563409/10400000\) | \(42598400000\) | \([2]\) | \(1920\) | \(1.0501\) | \(\Gamma_0(N)\)-optimal |
1040.b2 | 1040e2 | \([0, 1, 0, -12176, 715540]\) | \(-48743122863889/26406250000\) | \(-108160000000000\) | \([2]\) | \(3840\) | \(1.3967\) |
Rank
sage: E.rank()
The elliptic curves in class 1040e have rank \(0\).
Complex multiplication
The elliptic curves in class 1040e do not have complex multiplication.Modular form 1040.2.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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.