Show commands:
SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 157170c
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 157170.da3 | 157170c1 | \([1, 0, 0, -41155, 3209777]\) | \(1597099875769/186000\) | \(897786474000\) | \([2]\) | \(663552\) | \(1.3199\) | \(\Gamma_0(N)\)-optimal |
| 157170.da2 | 157170c2 | \([1, 0, 0, -44535, 2650725]\) | \(2023804595449/540562500\) | \(2609191940062500\) | \([2, 2]\) | \(1327104\) | \(1.6665\) | |
| 157170.da4 | 157170c3 | \([1, 0, 0, 112635, 17204667]\) | \(32740359775271/45410156250\) | \(-219186150878906250\) | \([2]\) | \(2654208\) | \(2.0131\) | |
| 157170.da1 | 157170c4 | \([1, 0, 0, -255785, -47669025]\) | \(383432500775449/18701300250\) | \(90267604358402250\) | \([2]\) | \(2654208\) | \(2.0131\) |
Rank
sage: E.rank()
The elliptic curves in class 157170c have rank \(1\).
Complex multiplication
The elliptic curves in class 157170c do not have complex multiplication.Modular form 157170.2.a.c
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
