Show commands for:
SageMath
sage: E = EllipticCurve("h1")
sage: E.isogeny_class()
Elliptic curves in class 630.h
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | Torsion structure | Modular degree | Optimality |
---|---|---|---|---|---|
630.h1 | 630i7 | [1, -1, 1, -3161093, 2164026557] | [6] | 9216 | |
630.h2 | 630i6 | [1, -1, 1, -197573, 33848381] | [2, 6] | 4608 | |
630.h3 | 630i8 | [1, -1, 1, -183173, 38980541] | [6] | 9216 | |
630.h4 | 630i4 | [1, -1, 1, -39218, 2946557] | [2] | 3072 | |
630.h5 | 630i3 | [1, -1, 1, -13253, 449597] | [6] | 2304 | |
630.h6 | 630i2 | [1, -1, 1, -5198, -74419] | [2, 2] | 1536 | |
630.h7 | 630i1 | [1, -1, 1, -4478, -114163] | [2] | 768 | \(\Gamma_0(N)\)-optimal |
630.h8 | 630i5 | [1, -1, 1, 17302, -560419] | [2] | 3072 |
Rank
sage: E.rank()
The elliptic curves in class 630.h have rank \(0\).
Complex multiplication
The elliptic curves in class 630.h do not have complex multiplication.Modular form 630.2.a.h
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 3 & 4 & 6 & 12 & 12 \\ 2 & 1 & 2 & 6 & 2 & 3 & 6 & 6 \\ 4 & 2 & 1 & 12 & 4 & 6 & 12 & 3 \\ 3 & 6 & 12 & 1 & 12 & 2 & 4 & 4 \\ 4 & 2 & 4 & 12 & 1 & 6 & 3 & 12 \\ 6 & 3 & 6 & 2 & 6 & 1 & 2 & 2 \\ 12 & 6 & 12 & 4 & 3 & 2 & 1 & 4 \\ 12 & 6 & 3 & 4 & 12 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.