Show commands for:
SageMath
sage: E = EllipticCurve("30030.p1")
sage: E.isogeny_class()
Elliptic curves in class 30030.p
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | Torsion structure | Modular degree | Optimality |
---|---|---|---|---|---|
30030.p1 | 30030r8 | [1, 0, 1, -105752574, -418583571128] | [2] | 5971968 | |
30030.p2 | 30030r6 | [1, 0, 1, -6887574, -5960607128] | [2, 2] | 2985984 | |
30030.p3 | 30030r5 | [1, 0, 1, -2313909, 426827206] | [6] | 1990656 | |
30030.p4 | 30030r3 | [1, 0, 1, -1887574, 907392872] | [2] | 1492992 | |
30030.p5 | 30030r2 | [1, 0, 1, -1840059, 959624146] | [2, 6] | 995328 | |
30030.p6 | 30030r1 | [1, 0, 1, -1839559, 960172346] | [6] | 497664 | \(\Gamma_0(N)\)-optimal |
30030.p7 | 30030r4 | [1, 0, 1, -1374209, 1457338286] | [6] | 1990656 | |
30030.p8 | 30030r7 | [1, 0, 1, 11977426, -32869643128] | [2] | 5971968 |
Rank
sage: E.rank()
The elliptic curves in class 30030.p have rank \(1\).
Modular form 30030.2.a.p
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 6 & 12 & 12 & 4 \\ 2 & 1 & 6 & 2 & 3 & 6 & 6 & 2 \\ 3 & 6 & 1 & 12 & 2 & 4 & 4 & 12 \\ 4 & 2 & 12 & 1 & 6 & 3 & 12 & 4 \\ 6 & 3 & 2 & 6 & 1 & 2 & 2 & 6 \\ 12 & 6 & 4 & 3 & 2 & 1 & 4 & 12 \\ 12 & 6 & 4 & 12 & 2 & 4 & 1 & 3 \\ 4 & 2 & 12 & 4 & 6 & 12 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.