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SageMath
E = EllipticCurve("ba1")
E.isogeny_class()
Elliptic curves in class 8880.ba
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 8880.ba1 | 8880ba3 | \([0, 1, 0, -5456080, -4907160172]\) | \(4385367890843575421521/24975000000\) | \(102297600000000\) | \([2]\) | \(165888\) | \(2.3000\) | |
| 8880.ba2 | 8880ba5 | \([0, 1, 0, -4850000, 4091635860]\) | \(3080272010107543650001/15465841417699560\) | \(63348086446897397760\) | \([4]\) | \(331776\) | \(2.6465\) | |
| 8880.ba3 | 8880ba4 | \([0, 1, 0, -469200, -14049900]\) | \(2788936974993502801/1593609593601600\) | \(6527424895392153600\) | \([2, 4]\) | \(165888\) | \(2.3000\) | |
| 8880.ba4 | 8880ba2 | \([0, 1, 0, -341200, -76667500]\) | \(1072487167529950801/2554882560000\) | \(10464798965760000\) | \([2, 2]\) | \(82944\) | \(1.9534\) | |
| 8880.ba5 | 8880ba1 | \([0, 1, 0, -13520, -2087532]\) | \(-66730743078481/419010969600\) | \(-1716268931481600\) | \([2]\) | \(41472\) | \(1.6068\) | \(\Gamma_0(N)\)-optimal |
| 8880.ba6 | 8880ba6 | \([0, 1, 0, 1863600, -110161260]\) | \(174751791402194852399/102423900876336360\) | \(-419528297989473730560\) | \([4]\) | \(331776\) | \(2.6465\) |
Rank
sage: E.rank()
The elliptic curves in class 8880.ba have rank \(1\).
Complex multiplication
The elliptic curves in class 8880.ba do not have complex multiplication.Modular form 8880.2.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 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.
