Show commands:
SageMath
E = EllipticCurve("x1")
E.isogeny_class()
Elliptic curves in class 187200.x
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
187200.x1 | 187200kv4 | \([0, 0, 0, -1246010700, -16928979194000]\) | \(71647584155243142409/10140000\) | \(30277877760000000000\) | \([2]\) | \(47185920\) | \(3.5910\) | |
187200.x2 | 187200kv3 | \([0, 0, 0, -89402700, -181083386000]\) | \(26465989780414729/10571870144160\) | \(31567435100539453440000000\) | \([2]\) | \(47185920\) | \(3.5910\) | |
187200.x3 | 187200kv2 | \([0, 0, 0, -77882700, -264465146000]\) | \(17496824387403529/6580454400\) | \(19649131551129600000000\) | \([2, 2]\) | \(23592960\) | \(3.2444\) | |
187200.x4 | 187200kv1 | \([0, 0, 0, -4154700, -5384954000]\) | \(-2656166199049/2658140160\) | \(-7937163987517440000000\) | \([2]\) | \(11796480\) | \(2.8979\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 187200.x have rank \(0\).
Complex multiplication
The elliptic curves in class 187200.x do not have complex multiplication.Modular form 187200.2.a.x
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.