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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 178a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
178.b2 | 178a1 | \([1, 0, 0, 6, -28]\) | \(23639903/364544\) | \(-364544\) | \([3]\) | \(32\) | \(-0.25219\) | \(\Gamma_0(N)\)-optimal |
178.b1 | 178a2 | \([1, 0, 0, -554, -5068]\) | \(-18806241149857/11279504\) | \(-11279504\) | \([]\) | \(96\) | \(0.29712\) |
Rank
sage: E.rank()
The elliptic curves in class 178a have rank \(0\).
Complex multiplication
The elliptic curves in class 178a do not have complex multiplication.Modular form 178.2.a.a
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.