Show commands:
SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 30a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
30.a8 | 30a1 | \([1, 0, 1, 1, 2]\) | \(357911/2160\) | \(-2160\) | \([6]\) | \(2\) | \(-0.67829\) | \(\Gamma_0(N)\)-optimal |
30.a6 | 30a2 | \([1, 0, 1, -19, 26]\) | \(702595369/72900\) | \(72900\) | \([2, 6]\) | \(4\) | \(-0.33171\) | |
30.a7 | 30a3 | \([1, 0, 1, -14, -64]\) | \(-273359449/1536000\) | \(-1536000\) | \([2]\) | \(6\) | \(-0.12898\) | |
30.a5 | 30a4 | \([1, 0, 1, -69, -194]\) | \(35578826569/5314410\) | \(5314410\) | \([6]\) | \(8\) | \(0.014860\) | |
30.a4 | 30a5 | \([1, 0, 1, -289, 1862]\) | \(2656166199049/33750\) | \(33750\) | \([6]\) | \(8\) | \(0.014860\) | |
30.a3 | 30a6 | \([1, 0, 1, -334, -2368]\) | \(4102915888729/9000000\) | \(9000000\) | \([2, 2]\) | \(12\) | \(0.21759\) | |
30.a1 | 30a7 | \([1, 0, 1, -5334, -150368]\) | \(16778985534208729/81000\) | \(81000\) | \([2]\) | \(24\) | \(0.56417\) | |
30.a2 | 30a8 | \([1, 0, 1, -454, -544]\) | \(10316097499609/5859375000\) | \(5859375000\) | \([2]\) | \(24\) | \(0.56417\) |
Rank
sage: E.rank()
The elliptic curves in class 30a have rank \(0\).
Complex multiplication
The elliptic curves in class 30a do not have complex multiplication.Modular form 30.2.a.a
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.