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SageMath
E = EllipticCurve("hl1")
E.isogeny_class()
Elliptic curves in class 78400.hl
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 78400.hl1 | 78400hm4 | \([0, 1, 0, -9840833, -11885689537]\) | \(-349938025/8\) | \(-2409451520000000000\) | \([]\) | \(2073600\) | \(2.6406\) | |
| 78400.hl2 | 78400hm3 | \([0, 1, 0, -40833, -37489537]\) | \(-25/2\) | \(-602362880000000000\) | \([]\) | \(691200\) | \(2.0913\) | |
| 78400.hl3 | 78400hm1 | \([0, 1, 0, -9473, 424703]\) | \(-121945/32\) | \(-24672783564800\) | \([]\) | \(138240\) | \(1.2866\) | \(\Gamma_0(N)\)-optimal |
| 78400.hl4 | 78400hm2 | \([0, 1, 0, 68927, -3134657]\) | \(46969655/32768\) | \(-25264930370355200\) | \([]\) | \(414720\) | \(1.8359\) |
Rank
sage: E.rank()
The elliptic curves in class 78400.hl have rank \(1\).
Complex multiplication
The elliptic curves in class 78400.hl do not have complex multiplication.Modular form 78400.2.a.hl
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.
\(\left(\begin{array}{rrrr} 1 & 3 & 15 & 5 \\ 3 & 1 & 5 & 15 \\ 15 & 5 & 1 & 3 \\ 5 & 15 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
