Show commands:
SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 5850.c
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
5850.c1 | 5850r3 | \([1, -1, 0, -19468917, -33059545259]\) | \(71647584155243142409/10140000\) | \(115500937500000\) | \([2]\) | \(245760\) | \(2.5513\) | |
5850.c2 | 5850r4 | \([1, -1, 0, -1396917, -353329259]\) | \(26465989780414729/10571870144160\) | \(120420208360822500000\) | \([2]\) | \(245760\) | \(2.5513\) | |
5850.c3 | 5850r2 | \([1, -1, 0, -1216917, -516229259]\) | \(17496824387403529/6580454400\) | \(74955488400000000\) | \([2, 2]\) | \(122880\) | \(2.2047\) | |
5850.c4 | 5850r1 | \([1, -1, 0, -64917, -10501259]\) | \(-2656166199049/2658140160\) | \(-30277877760000000\) | \([2]\) | \(61440\) | \(1.8581\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 5850.c have rank \(1\).
Complex multiplication
The elliptic curves in class 5850.c do not have complex multiplication.Modular form 5850.2.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 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.