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SageMath
E = EllipticCurve("q1")
E.isogeny_class()
Elliptic curves in class 51870.q
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 51870.q1 | 51870l4 | \([1, 1, 0, -5901422, 5429579034]\) | \(22729707196852465027832041/406726014705883904850\) | \(406726014705883904850\) | \([2]\) | \(3538944\) | \(2.7503\) | |
| 51870.q2 | 51870l2 | \([1, 1, 0, -757172, -125182116]\) | \(48007406511374545940041/21046460496456622500\) | \(21046460496456622500\) | \([2, 2]\) | \(1769472\) | \(2.4037\) | |
| 51870.q3 | 51870l1 | \([1, 1, 0, -644672, -199409616]\) | \(29630650678461777740041/15483292256250000\) | \(15483292256250000\) | \([2]\) | \(884736\) | \(2.0571\) | \(\Gamma_0(N)\)-optimal |
| 51870.q4 | 51870l3 | \([1, 1, 0, 2587078, -927133266]\) | \(1914926099034582908751959/1480375683617401784850\) | \(-1480375683617401784850\) | \([2]\) | \(3538944\) | \(2.7503\) |
Rank
sage: E.rank()
The elliptic curves in class 51870.q have rank \(0\).
Complex multiplication
The elliptic curves in class 51870.q do not have complex multiplication.Modular form 51870.2.a.q
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
