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SageMath
E = EllipticCurve("ha1")
E.isogeny_class()
Elliptic curves in class 84150.ha
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 84150.ha1 | 84150fq4 | \([1, -1, 1, -40145855, 85227271647]\) | \(628200507126935410849/88124751829125000\) | \(1003796001303626953125000\) | \([2]\) | \(15925248\) | \(3.3317\) | |
| 84150.ha2 | 84150fq2 | \([1, -1, 1, -10256855, -12626398353]\) | \(10476561483361670689/13992628953600\) | \(159384789174600000000\) | \([2]\) | \(5308416\) | \(2.7824\) | |
| 84150.ha3 | 84150fq1 | \([1, -1, 1, -464855, -308062353]\) | \(-975276594443809/3037581803520\) | \(-34599955230720000000\) | \([2]\) | \(2654208\) | \(2.4358\) | \(\Gamma_0(N)\)-optimal |
| 84150.ha4 | 84150fq3 | \([1, -1, 1, 4071145, 7140049647]\) | \(655127711084516831/2313151512408000\) | \(-26348241446022375000000\) | \([2]\) | \(7962624\) | \(2.9851\) |
Rank
sage: E.rank()
The elliptic curves in class 84150.ha have rank \(0\).
Complex multiplication
The elliptic curves in class 84150.ha do not have complex multiplication.Modular form 84150.2.a.ha
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.
