Show commands:
SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 8400c
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
8400.l4 | 8400c1 | \([0, -1, 0, -4383, -110238]\) | \(37256083456/525\) | \(131250000\) | \([2]\) | \(6144\) | \(0.69791\) | \(\Gamma_0(N)\)-optimal |
8400.l3 | 8400c2 | \([0, -1, 0, -4508, -103488]\) | \(2533446736/275625\) | \(1102500000000\) | \([2, 2]\) | \(12288\) | \(1.0445\) | |
8400.l2 | 8400c3 | \([0, -1, 0, -17008, 746512]\) | \(34008619684/4862025\) | \(77792400000000\) | \([2, 2]\) | \(24576\) | \(1.3911\) | |
8400.l5 | 8400c4 | \([0, -1, 0, 5992, -523488]\) | \(1486779836/8203125\) | \(-131250000000000\) | \([2]\) | \(24576\) | \(1.3911\) | |
8400.l1 | 8400c5 | \([0, -1, 0, -262008, 51706512]\) | \(62161150998242/1607445\) | \(51438240000000\) | \([2]\) | \(49152\) | \(1.7376\) | |
8400.l6 | 8400c6 | \([0, -1, 0, 27992, 3986512]\) | \(75798394558/259416045\) | \(-8301313440000000\) | \([2]\) | \(49152\) | \(1.7376\) |
Rank
sage: E.rank()
The elliptic curves in class 8400c have rank \(1\).
Complex multiplication
The elliptic curves in class 8400c do not have complex multiplication.Modular form 8400.2.a.c
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.