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SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 2898i
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 2898.i5 | 2898i1 | \([1, -1, 0, 1134, -30380]\) | \(221115865823/664731648\) | \(-484589371392\) | \([2]\) | \(4096\) | \(0.92556\) | \(\Gamma_0(N)\)-optimal |
| 2898.i4 | 2898i2 | \([1, -1, 0, -10386, -346028]\) | \(169967019783457/26337394944\) | \(19199960914176\) | \([2, 2]\) | \(8192\) | \(1.2721\) | |
| 2898.i2 | 2898i3 | \([1, -1, 0, -159426, -24460700]\) | \(614716917569296417/19093020912\) | \(13918812244848\) | \([2]\) | \(16384\) | \(1.6187\) | |
| 2898.i3 | 2898i4 | \([1, -1, 0, -45666, 3428932]\) | \(14447092394873377/1439452851984\) | \(1049361129096336\) | \([2, 2]\) | \(16384\) | \(1.6187\) | |
| 2898.i1 | 2898i5 | \([1, -1, 0, -712206, 231518920]\) | \(54804145548726848737/637608031452\) | \(464816254928508\) | \([2]\) | \(32768\) | \(1.9653\) | |
| 2898.i6 | 2898i6 | \([1, -1, 0, 56394, 16513024]\) | \(27207619911317663/177609314617308\) | \(-129477190356017532\) | \([2]\) | \(32768\) | \(1.9653\) |
Rank
sage: E.rank()
The elliptic curves in class 2898i have rank \(0\).
Complex multiplication
The elliptic curves in class 2898i do not have complex multiplication.Modular form 2898.2.a.i
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 Cremona numbering.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
