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SageMath
E = EllipticCurve("h1")
E.isogeny_class()
Elliptic curves in class 17661.h
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 17661.h1 | 17661f4 | \([1, 0, 1, -172088802, 868900127401]\) | \(947531277805646290177/38367\) | \(22821586356807\) | \([2]\) | \(1290240\) | \(2.9743\) | |
| 17661.h2 | 17661f5 | \([1, 0, 1, -35716447, -66790896769]\) | \(8471112631466271697/1662662681263647\) | \(988990537772006985111687\) | \([2]\) | \(2580480\) | \(3.3209\) | |
| 17661.h3 | 17661f3 | \([1, 0, 1, -10961612, 13028593205]\) | \(244883173420511137/18418027974129\) | \(10955472565842313862409\) | \([2, 2]\) | \(1290240\) | \(2.9743\) | |
| 17661.h4 | 17661f2 | \([1, 0, 1, -10755567, 13575848725]\) | \(231331938231569617/1472026689\) | \(875595803751614169\) | \([2, 2]\) | \(645120\) | \(2.6277\) | |
| 17661.h5 | 17661f1 | \([1, 0, 1, -659362, 220588751]\) | \(-53297461115137/4513839183\) | \(-2684936813291986743\) | \([2]\) | \(322560\) | \(2.2812\) | \(\Gamma_0(N)\)-optimal |
| 17661.h6 | 17661f6 | \([1, 0, 1, 10496503, 57824554079]\) | \(215015459663151503/2552757445339983\) | \(-1518439661344604662143543\) | \([2]\) | \(2580480\) | \(3.3209\) |
Rank
sage: E.rank()
The elliptic curves in class 17661.h have rank \(1\).
Complex multiplication
The elliptic curves in class 17661.h do not have complex multiplication.Modular form 17661.2.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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 4 & 2 & 4 & 8 \\ 8 & 1 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 2 & 4 & 2 & 1 & 2 & 4 \\ 4 & 8 & 4 & 2 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
