Show commands:
SageMath
E = EllipticCurve("h1")
E.isogeny_class()
Elliptic curves in class 8281.h
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
8281.h1 | 8281d3 | \([0, -1, 1, -971637, 908886292]\) | \(-178643795968/524596891\) | \(-297902444115204004531\) | \([]\) | \(290304\) | \(2.6155\) | |
8281.h2 | 8281d1 | \([0, -1, 1, -60727, -5750158]\) | \(-43614208/91\) | \(-51676101935731\) | \([]\) | \(32256\) | \(1.5169\) | \(\Gamma_0(N)\)-optimal |
8281.h3 | 8281d2 | \([0, -1, 1, 104893, -28663685]\) | \(224755712/753571\) | \(-427929800129788411\) | \([]\) | \(96768\) | \(2.0662\) |
Rank
sage: E.rank()
The elliptic curves in class 8281.h have rank \(0\).
Complex multiplication
The elliptic curves in class 8281.h do not have complex multiplication.Modular form 8281.2.a.h
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.
\(\left(\begin{array}{rrr} 1 & 9 & 3 \\ 9 & 1 & 3 \\ 3 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.