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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 19.a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
19.a1 | 19a2 | \([0, 1, 1, -769, -8470]\) | \(-50357871050752/19\) | \(-19\) | \([]\) | \(3\) | \(0.033439\) | |
19.a2 | 19a1 | \([0, 1, 1, -9, -15]\) | \(-89915392/6859\) | \(-6859\) | \([3]\) | \(1\) | \(-0.51587\) | \(\Gamma_0(N)\)-optimal |
19.a3 | 19a3 | \([0, 1, 1, 1, 0]\) | \(32768/19\) | \(-19\) | \([3]\) | \(3\) | \(-1.0652\) |
Rank
sage: E.rank()
The elliptic curves in class 19.a have rank \(0\).
Complex multiplication
The elliptic curves in class 19.a do not have complex multiplication.Modular form 19.2.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 LMFDB 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 LMFDB labels.