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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 4624.a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
4624.a1 | 4624f4 | \([0, 1, 0, -522608, 100312660]\) | \(159661140625/48275138\) | \(4772841367386202112\) | \([2]\) | \(82944\) | \(2.2892\) | |
4624.a2 | 4624f3 | \([0, 1, 0, -476368, 126373524]\) | \(120920208625/19652\) | \(1942943768526848\) | \([2]\) | \(41472\) | \(1.9427\) | |
4624.a3 | 4624f2 | \([0, 1, 0, -198928, -34208748]\) | \(8805624625/2312\) | \(228581619826688\) | \([2]\) | \(27648\) | \(1.7399\) | |
4624.a4 | 4624f1 | \([0, 1, 0, -13968, -398060]\) | \(3048625/1088\) | \(107567821094912\) | \([2]\) | \(13824\) | \(1.3934\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 4624.a have rank \(0\).
Complex multiplication
The elliptic curves in class 4624.a do not have complex multiplication.Modular form 4624.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.