Show commands:
SageMath
E = EllipticCurve("co1")
E.isogeny_class()
Elliptic curves in class 12480co
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
12480.cn4 | 12480co1 | \([0, 1, 0, 13044, -403506]\) | \(3834800837445824/3342041015625\) | \(-213890625000000\) | \([2]\) | \(49152\) | \(1.4372\) | \(\Gamma_0(N)\)-optimal |
12480.cn3 | 12480co2 | \([0, 1, 0, -65081, -3637881]\) | \(7442744143086784/2927948765625\) | \(11992878144000000\) | \([2, 2]\) | \(98304\) | \(1.7838\) | |
12480.cn1 | 12480co3 | \([0, 1, 0, -910081, -334370881]\) | \(2543984126301795848/909361981125\) | \(29797973397504000\) | \([2]\) | \(196608\) | \(2.1304\) | |
12480.cn2 | 12480co4 | \([0, 1, 0, -470081, 121345119]\) | \(350584567631475848/8259273550125\) | \(270639875690496000\) | \([2]\) | \(196608\) | \(2.1304\) |
Rank
sage: E.rank()
The elliptic curves in class 12480co have rank \(1\).
Complex multiplication
The elliptic curves in class 12480co do not have complex multiplication.Modular form 12480.2.a.co
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.