Show commands:
SageMath
E = EllipticCurve("w1")
E.isogeny_class()
Elliptic curves in class 8880.w
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 8880.w1 | 8880x3 | \([0, 1, 0, -5733089736, 167080688257140]\) | \(5087799435928552778197163696329/125914832087040\) | \(515747152228515840\) | \([2]\) | \(4730880\) | \(3.8443\) | |
| 8880.w2 | 8880x2 | \([0, 1, 0, -358318536, 2610539628660]\) | \(1242142983306846366056931529/6179359141291622400\) | \(25310655042730485350400\) | \([2, 2]\) | \(2365440\) | \(3.4978\) | |
| 8880.w3 | 8880x4 | \([0, 1, 0, -352256456, 2703136688244]\) | \(-1180159344892952613848670409/87759036144023189760000\) | \(-359461012045918985256960000\) | \([2]\) | \(4730880\) | \(3.8443\) | |
| 8880.w4 | 8880x1 | \([0, 1, 0, -22774216, 39330613364]\) | \(318929057401476905525449/21353131537921474560\) | \(87462426779326359797760\) | \([2]\) | \(1182720\) | \(3.1512\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 8880.w have rank \(1\).
Complex multiplication
The elliptic curves in class 8880.w do not have complex multiplication.Modular form 8880.2.a.w
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
