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SageMath
E = EllipticCurve("bh1")
E.isogeny_class()
Elliptic curves in class 121680bh
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
121680.er5 | 121680bh1 | \([0, 0, 0, -23322, 1368731]\) | \(24918016/45\) | \(2533495507920\) | \([2]\) | \(245760\) | \(1.2721\) | \(\Gamma_0(N)\)-optimal |
121680.er4 | 121680bh2 | \([0, 0, 0, -30927, 399854]\) | \(3631696/2025\) | \(1824116765702400\) | \([2, 2]\) | \(491520\) | \(1.6186\) | |
121680.er6 | 121680bh3 | \([0, 0, 0, 121173, 3168074]\) | \(54607676/32805\) | \(-118202766417515520\) | \([2]\) | \(983040\) | \(1.9652\) | |
121680.er2 | 121680bh4 | \([0, 0, 0, -304707, -64376494]\) | \(868327204/5625\) | \(20267964063360000\) | \([2, 2]\) | \(983040\) | \(1.9652\) | |
121680.er3 | 121680bh5 | \([0, 0, 0, -122187, -140779366]\) | \(-27995042/1171875\) | \(-8444985026400000000\) | \([2]\) | \(1966080\) | \(2.3118\) | |
121680.er1 | 121680bh6 | \([0, 0, 0, -4867707, -4133659894]\) | \(1770025017602/75\) | \(540479041689600\) | \([2]\) | \(1966080\) | \(2.3118\) |
Rank
sage: E.rank()
The elliptic curves in class 121680bh have rank \(1\).
Complex multiplication
The elliptic curves in class 121680bh do not have complex multiplication.Modular form 121680.2.a.bh
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.