Show commands:
SageMath
E = EllipticCurve("bt1")
E.isogeny_class()
Elliptic curves in class 352800.bt
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 352800.bt1 | 352800bt4 | \([0, 0, 0, -926103675, -10847696476750]\) | \(128025588102048008/7875\) | \(5403265623000000000\) | \([2]\) | \(56623104\) | \(3.5029\) | |
| 352800.bt2 | 352800bt2 | \([0, 0, 0, -64830675, -126250668250]\) | \(43919722445768/15380859375\) | \(10553253169921875000000000\) | \([2]\) | \(56623104\) | \(3.5029\) | |
| 352800.bt3 | 352800bt1 | \([0, 0, 0, -57884925, -169474070500]\) | \(250094631024064/62015625\) | \(5318839597640625000000\) | \([2, 2]\) | \(28311552\) | \(3.1564\) | \(\Gamma_0(N)\)-optimal |
| 352800.bt4 | 352800bt3 | \([0, 0, 0, -50994300, -211341508000]\) | \(-2671731885376/1969120125\) | \(-10808562873874248000000000\) | \([2]\) | \(56623104\) | \(3.5029\) |
Rank
sage: E.rank()
The elliptic curves in class 352800.bt have rank \(1\).
Complex multiplication
The elliptic curves in class 352800.bt do not have complex multiplication.Modular form 352800.2.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 LMFDB numbering.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
