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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 6690.a
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 6690.a1 | 6690a3 | \([1, 1, 0, -84188593, 288828342613]\) | \(65990812340278189764070806169/2142150634958771454912000\) | \(2142150634958771454912000\) | \([2]\) | \(1384128\) | \(3.4424\) | |
| 6690.a2 | 6690a2 | \([1, 1, 0, -12828593, -11411721387]\) | \(233486000400975208694166169/78913673205682176000000\) | \(78913673205682176000000\) | \([2, 2]\) | \(692064\) | \(3.0958\) | |
| 6690.a3 | 6690a1 | \([1, 1, 0, -11517873, -15047396523]\) | \(168982070711351853939176089/37703877214076928000\) | \(37703877214076928000\) | \([2]\) | \(346032\) | \(2.7493\) | \(\Gamma_0(N)\)-optimal |
| 6690.a4 | 6690a4 | \([1, 1, 0, 37559887, -78942362283]\) | \(5859985279907178462243106151/6084442029900375000000000\) | \(-6084442029900375000000000\) | \([2]\) | \(1384128\) | \(3.4424\) |
Rank
sage: E.rank()
The elliptic curves in class 6690.a have rank \(1\).
Complex multiplication
The elliptic curves in class 6690.a do not have complex multiplication.Modular form 6690.2.a.a
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.
