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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 98.a
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 98.a1 | 98a6 | \([1, 1, 0, -133795, 18781197]\) | \(2251439055699625/25088\) | \(2951578112\) | \([2]\) | \(288\) | \(1.3861\) | |
| 98.a2 | 98a5 | \([1, 1, 0, -8355, 291341]\) | \(-548347731625/1835008\) | \(-215886856192\) | \([2]\) | \(144\) | \(1.0395\) | |
| 98.a3 | 98a4 | \([1, 1, 0, -1740, 22184]\) | \(4956477625/941192\) | \(110730297608\) | \([2]\) | \(96\) | \(0.83675\) | |
| 98.a4 | 98a2 | \([1, 1, 0, -515, -4717]\) | \(128787625/98\) | \(11529602\) | \([2]\) | \(32\) | \(0.28744\) | |
| 98.a5 | 98a1 | \([1, 1, 0, -25, -111]\) | \(-15625/28\) | \(-3294172\) | \([2]\) | \(16\) | \(-0.059130\) | \(\Gamma_0(N)\)-optimal |
| 98.a6 | 98a3 | \([1, 1, 0, 220, 2192]\) | \(9938375/21952\) | \(-2582630848\) | \([2]\) | \(48\) | \(0.49018\) |
Rank
sage: E.rank()
The elliptic curves in class 98.a have rank \(0\).
Complex multiplication
The elliptic curves in class 98.a do not have complex multiplication.Modular form 98.2.a.a
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 3 & 9 & 18 & 6 \\ 2 & 1 & 6 & 18 & 9 & 3 \\ 3 & 6 & 1 & 3 & 6 & 2 \\ 9 & 18 & 3 & 1 & 2 & 6 \\ 18 & 9 & 6 & 2 & 1 & 3 \\ 6 & 3 & 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
