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SageMath
E = EllipticCurve("bt1")
E.isogeny_class()
Elliptic curves in class 131648.bt
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
131648.bt1 | 131648ch3 | \([0, 0, 0, -702284, 226525552]\) | \(82483294977/17\) | \(7894869475328\) | \([2]\) | \(737280\) | \(1.8620\) | |
131648.bt2 | 131648ch2 | \([0, 0, 0, -44044, 3513840]\) | \(20346417/289\) | \(134212781080576\) | \([2, 2]\) | \(368640\) | \(1.5155\) | |
131648.bt3 | 131648ch1 | \([0, 0, 0, -5324, -63888]\) | \(35937/17\) | \(7894869475328\) | \([2]\) | \(184320\) | \(1.1689\) | \(\Gamma_0(N)\)-optimal |
131648.bt4 | 131648ch4 | \([0, 0, 0, -5324, 9476720]\) | \(-35937/83521\) | \(-38787493732286464\) | \([2]\) | \(737280\) | \(1.8620\) |
Rank
sage: E.rank()
The elliptic curves in class 131648.bt have rank \(0\).
Complex multiplication
The elliptic curves in class 131648.bt do not have complex multiplication.Modular form 131648.2.a.bt
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.