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SageMath
sage: E = EllipticCurve("k1")
sage: E.isogeny_class()
Elliptic curves in class 303450.k
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 303450.k1 | 303450k3 | \([1, 1, 0, -36695925, 85545487125]\) | \(14489843500598257/6246072\) | \(2355703029358875000\) | \([2]\) | \(28311552\) | \(2.8679\) | |
| 303450.k2 | 303450k4 | \([1, 1, 0, -4905925, -2212830875]\) | \(34623662831857/14438442312\) | \(5445451524350305125000\) | \([2]\) | \(28311552\) | \(2.8679\) | |
| 303450.k3 | 303450k2 | \([1, 1, 0, -2304925, 1321928125]\) | \(3590714269297/73410624\) | \(27686781283329000000\) | \([2, 2]\) | \(14155776\) | \(2.5213\) | |
| 303450.k4 | 303450k1 | \([1, 1, 0, 7075, 61888125]\) | \(103823/4386816\) | \(-1654485529536000000\) | \([2]\) | \(7077888\) | \(2.1747\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 303450.k have rank \(1\).
Complex multiplication
The elliptic curves in class 303450.k do not have complex multiplication.Modular form 303450.2.a.k
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.
