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SageMath
E = EllipticCurve("hl1")
E.isogeny_class()
Elliptic curves in class 129360.hl
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
129360.hl1 | 129360hz4 | \([0, 1, 0, -2430220, 768028568]\) | \(52702650535889104/22020583921875\) | \(663219117523116000000\) | \([2]\) | \(5971968\) | \(2.6924\) | |
129360.hl2 | 129360hz2 | \([0, 1, 0, -2095060, 1166495000]\) | \(33766427105425744/9823275\) | \(295858811001600\) | \([2]\) | \(1990656\) | \(2.1431\) | |
129360.hl3 | 129360hz1 | \([0, 1, 0, -130405, 18350618]\) | \(-130287139815424/2250652635\) | \(-4236592509681840\) | \([2]\) | \(995328\) | \(1.7965\) | \(\Gamma_0(N)\)-optimal |
129360.hl4 | 129360hz3 | \([0, 1, 0, 504635, 88316150]\) | \(7549996227362816/6152409907875\) | \(-11581197972025374000\) | \([2]\) | \(2985984\) | \(2.3458\) |
Rank
sage: E.rank()
The elliptic curves in class 129360.hl have rank \(0\).
Complex multiplication
The elliptic curves in class 129360.hl do not have complex multiplication.Modular form 129360.2.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.
\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.