Show commands:
SageMath
E = EllipticCurve("cr1")
E.isogeny_class()
Elliptic curves in class 54150.cr
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 54150.cr1 | 54150cv4 | \([1, 0, 0, -7472888, -7837888608]\) | \(502270291349/1889568\) | \(173625764198062500000\) | \([2]\) | \(2880000\) | \(2.7430\) | |
| 54150.cr2 | 54150cv2 | \([1, 0, 0, -478513, 127350767]\) | \(131872229/18\) | \(1653956753906250\) | \([2]\) | \(576000\) | \(1.9382\) | |
| 54150.cr3 | 54150cv3 | \([1, 0, 0, -252888, -235228608]\) | \(-19465109/248832\) | \(-22864298166000000000\) | \([2]\) | \(1440000\) | \(2.3964\) | |
| 54150.cr4 | 54150cv1 | \([1, 0, 0, -27263, 2354517]\) | \(-24389/12\) | \(-1102637835937500\) | \([2]\) | \(288000\) | \(1.5917\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 54150.cr have rank \(1\).
Complex multiplication
The elliptic curves in class 54150.cr do not have complex multiplication.Modular form 54150.2.a.cr
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.
\(\left(\begin{array}{rrrr} 1 & 5 & 2 & 10 \\ 5 & 1 & 10 & 2 \\ 2 & 10 & 1 & 5 \\ 10 & 2 & 5 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
