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SageMath
E = EllipticCurve("ba1")
E.isogeny_class()
Elliptic curves in class 11025ba
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 11025.g6 | 11025ba1 | \([1, -1, 1, 10795, -97828]\) | \(103823/63\) | \(-84426025359375\) | \([2]\) | \(24576\) | \(1.3615\) | \(\Gamma_0(N)\)-optimal |
| 11025.g5 | 11025ba2 | \([1, -1, 1, -44330, -759328]\) | \(7189057/3969\) | \(5318839597640625\) | \([2, 2]\) | \(49152\) | \(1.7080\) | |
| 11025.g2 | 11025ba3 | \([1, -1, 1, -540455, -152573578]\) | \(13027640977/21609\) | \(28958126698265625\) | \([2, 2]\) | \(98304\) | \(2.0546\) | |
| 11025.g3 | 11025ba4 | \([1, -1, 1, -430205, 108057422]\) | \(6570725617/45927\) | \(61546572486984375\) | \([2]\) | \(98304\) | \(2.0546\) | |
| 11025.g1 | 11025ba5 | \([1, -1, 1, -8643830, -9779383078]\) | \(53297461115137/147\) | \(196994059171875\) | \([2]\) | \(196608\) | \(2.4012\) | |
| 11025.g4 | 11025ba6 | \([1, -1, 1, -375080, -247829578]\) | \(-4354703137/17294403\) | \(-23176154067511921875\) | \([2]\) | \(196608\) | \(2.4012\) |
Rank
sage: E.rank()
The elliptic curves in class 11025ba have rank \(1\).
Complex multiplication
The elliptic curves in class 11025ba do not have complex multiplication.Modular form 11025.2.a.ba
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
