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SageMath
E = EllipticCurve("ew1")
E.isogeny_class()
Elliptic curves in class 215600ew
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 215600.hq4 | 215600ew1 | \([0, -1, 0, 195592, 272051312]\) | \(109902239/4312000\) | \(-32467359232000000000\) | \([2]\) | \(5308416\) | \(2.4238\) | \(\Gamma_0(N)\)-optimal |
| 215600.hq2 | 215600ew2 | \([0, -1, 0, -5292408, 4486835312]\) | \(2177286259681/105875000\) | \(797189624000000000000\) | \([2]\) | \(10616832\) | \(2.7703\) | |
| 215600.hq3 | 215600ew3 | \([0, -1, 0, -1764408, -7442508688]\) | \(-80677568161/3131816380\) | \(-23581124178599680000000\) | \([2]\) | \(15925248\) | \(2.9731\) | |
| 215600.hq1 | 215600ew4 | \([0, -1, 0, -68992408, -219345164688]\) | \(4823468134087681/30382271150\) | \(228764404385686400000000\) | \([2]\) | \(31850496\) | \(3.3196\) |
Rank
sage: E.rank()
The elliptic curves in class 215600ew have rank \(0\).
Complex multiplication
The elliptic curves in class 215600ew do not have complex multiplication.Modular form 215600.2.a.ew
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
