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SageMath
E = EllipticCurve("ba1")
E.isogeny_class()
Elliptic curves in class 53312.ba
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
53312.ba1 | 53312o4 | \([0, 0, 0, -284396, 58375856]\) | \(82483294977/17\) | \(524296650752\) | \([2]\) | \(147456\) | \(1.6360\) | |
53312.ba2 | 53312o2 | \([0, 0, 0, -17836, 905520]\) | \(20346417/289\) | \(8913043062784\) | \([2, 2]\) | \(73728\) | \(1.2895\) | |
53312.ba3 | 53312o1 | \([0, 0, 0, -2156, -16464]\) | \(35937/17\) | \(524296650752\) | \([2]\) | \(36864\) | \(0.94289\) | \(\Gamma_0(N)\)-optimal |
53312.ba4 | 53312o3 | \([0, 0, 0, -2156, 2442160]\) | \(-35937/83521\) | \(-2575869445144576\) | \([2]\) | \(147456\) | \(1.6360\) |
Rank
sage: E.rank()
The elliptic curves in class 53312.ba have rank \(2\).
Complex multiplication
The elliptic curves in class 53312.ba do not have complex multiplication.Modular form 53312.2.a.ba
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.