Show commands:
SageMath
E = EllipticCurve("bt1")
E.isogeny_class()
Elliptic curves in class 88725bt
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 88725.q5 | 88725bt1 | \([1, 0, 0, 633662, 1065511667]\) | \(373092501599/6718359375\) | \(-506691210882568359375\) | \([2]\) | \(3096576\) | \(2.6533\) | \(\Gamma_0(N)\)-optimal |
| 88725.q4 | 88725bt2 | \([1, 0, 0, -12569463, 16183089792]\) | \(2912015927948401/184878500625\) | \(13943331417550869140625\) | \([2, 2]\) | \(6193152\) | \(2.9999\) | |
| 88725.q3 | 88725bt3 | \([1, 0, 0, -38447588, -71931925833]\) | \(83339496416030401/18593645841225\) | \(1402312142019334391015625\) | \([2, 2]\) | \(12386304\) | \(3.3465\) | |
| 88725.q2 | 88725bt4 | \([1, 0, 0, -197941338, 1071875917917]\) | \(11372424889583066401/50586128775\) | \(3815149713223890234375\) | \([2]\) | \(12386304\) | \(3.3465\) | |
| 88725.q6 | 88725bt5 | \([1, 0, 0, 86506787, -442421647708]\) | \(949279533867428399/1670570708285115\) | \(-125992589529483869500546875\) | \([2]\) | \(24772608\) | \(3.6931\) | |
| 88725.q1 | 88725bt6 | \([1, 0, 0, -577451963, -5340699691458]\) | \(282352188585428161201/20813369346315\) | \(1569721226267458731796875\) | \([2]\) | \(24772608\) | \(3.6931\) |
Rank
sage: E.rank()
The elliptic curves in class 88725bt have rank \(1\).
Complex multiplication
The elliptic curves in class 88725bt do not have complex multiplication.Modular form 88725.2.a.bt
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.
