Show commands:
SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 14784.e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
14784.e1 | 14784j3 | \([0, -1, 0, -23969, 1428705]\) | \(46477380430664/286446699\) | \(9386285432832\) | \([2]\) | \(36864\) | \(1.3275\) | |
14784.e2 | 14784j2 | \([0, -1, 0, -2409, -7191]\) | \(377619516352/211789809\) | \(867491057664\) | \([2, 2]\) | \(18432\) | \(0.98092\) | |
14784.e3 | 14784j1 | \([0, -1, 0, -1804, -28850]\) | \(10150654719808/19370043\) | \(1239682752\) | \([2]\) | \(9216\) | \(0.63434\) | \(\Gamma_0(N)\)-optimal |
14784.e4 | 14784j4 | \([0, -1, 0, 9471, -66591]\) | \(2866919053816/1712145897\) | \(-56103596752896\) | \([2]\) | \(36864\) | \(1.3275\) |
Rank
sage: E.rank()
The elliptic curves in class 14784.e have rank \(0\).
Complex multiplication
The elliptic curves in class 14784.e do not have complex multiplication.Modular form 14784.2.a.e
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.