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SageMath
E = EllipticCurve("ha1")
E.isogeny_class()
Elliptic curves in class 91200.ha
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 91200.ha1 | 91200dn4 | \([0, 1, 0, -162433, 25143263]\) | \(115714886617/1539\) | \(6303744000000\) | \([2]\) | \(393216\) | \(1.5994\) | |
| 91200.ha2 | 91200dn2 | \([0, 1, 0, -10433, 367263]\) | \(30664297/3249\) | \(13307904000000\) | \([2, 2]\) | \(196608\) | \(1.2529\) | |
| 91200.ha3 | 91200dn1 | \([0, 1, 0, -2433, -40737]\) | \(389017/57\) | \(233472000000\) | \([2]\) | \(98304\) | \(0.90628\) | \(\Gamma_0(N)\)-optimal |
| 91200.ha4 | 91200dn3 | \([0, 1, 0, 13567, 1831263]\) | \(67419143/390963\) | \(-1601384448000000\) | \([2]\) | \(393216\) | \(1.5994\) |
Rank
sage: E.rank()
The elliptic curves in class 91200.ha have rank \(1\).
Complex multiplication
The elliptic curves in class 91200.ha do not have complex multiplication.Modular form 91200.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
