Show commands:
SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 960b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
960.c5 | 960b1 | \([0, -1, 0, -61, 205]\) | \(24918016/45\) | \(46080\) | \([2]\) | \(128\) | \(-0.21315\) | \(\Gamma_0(N)\)-optimal |
960.c4 | 960b2 | \([0, -1, 0, -81, 81]\) | \(3631696/2025\) | \(33177600\) | \([2, 2]\) | \(256\) | \(0.13343\) | |
960.c2 | 960b3 | \([0, -1, 0, -801, -8415]\) | \(868327204/5625\) | \(368640000\) | \([2, 2]\) | \(512\) | \(0.48000\) | |
960.c6 | 960b4 | \([0, -1, 0, 319, 321]\) | \(54607676/32805\) | \(-2149908480\) | \([2]\) | \(512\) | \(0.48000\) | |
960.c1 | 960b5 | \([0, -1, 0, -12801, -553215]\) | \(1770025017602/75\) | \(9830400\) | \([2]\) | \(1024\) | \(0.82657\) | |
960.c3 | 960b6 | \([0, -1, 0, -321, -18879]\) | \(-27995042/1171875\) | \(-153600000000\) | \([2]\) | \(1024\) | \(0.82657\) |
Rank
sage: E.rank()
The elliptic curves in class 960b have rank \(1\).
Complex multiplication
The elliptic curves in class 960b do not have complex multiplication.Modular form 960.2.a.b
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.