Show commands:
SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 510.c
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
510.c1 | 510d3 | \([1, 1, 1, -6541, -206341]\) | \(30949975477232209/478125000\) | \(478125000\) | \([2]\) | \(768\) | \(0.80053\) | |
510.c2 | 510d2 | \([1, 1, 1, -421, -3157]\) | \(8253429989329/936360000\) | \(936360000\) | \([2, 2]\) | \(384\) | \(0.45396\) | |
510.c3 | 510d1 | \([1, 1, 1, -101, 299]\) | \(114013572049/15667200\) | \(15667200\) | \([4]\) | \(192\) | \(0.10739\) | \(\Gamma_0(N)\)-optimal |
510.c4 | 510d4 | \([1, 1, 1, 579, -14757]\) | \(21464092074671/109596256200\) | \(-109596256200\) | \([2]\) | \(768\) | \(0.80053\) |
Rank
sage: E.rank()
The elliptic curves in class 510.c have rank \(1\).
Complex multiplication
The elliptic curves in class 510.c do not have complex multiplication.Modular form 510.2.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 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.