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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 231.a
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 231.a1 | 231a5 | \([1, 1, 1, -4519, -118810]\) | \(10206027697760497/5557167\) | \(5557167\) | \([2]\) | \(160\) | \(0.62279\) | |
| 231.a2 | 231a3 | \([1, 1, 1, -284, -1924]\) | \(2533811507137/58110129\) | \(58110129\) | \([2, 2]\) | \(80\) | \(0.27622\) | |
| 231.a3 | 231a2 | \([1, 1, 1, -39, 36]\) | \(6570725617/2614689\) | \(2614689\) | \([2, 4]\) | \(40\) | \(-0.070357\) | |
| 231.a4 | 231a1 | \([1, 1, 1, -34, 62]\) | \(4354703137/1617\) | \(1617\) | \([4]\) | \(20\) | \(-0.41693\) | \(\Gamma_0(N)\)-optimal |
| 231.a5 | 231a6 | \([1, 1, 1, 31, -5578]\) | \(3288008303/13504609503\) | \(-13504609503\) | \([2]\) | \(160\) | \(0.62279\) | |
| 231.a6 | 231a4 | \([1, 1, 1, 126, 432]\) | \(221115865823/190238433\) | \(-190238433\) | \([4]\) | \(80\) | \(0.27622\) |
Rank
sage: E.rank()
The elliptic curves in class 231.a have rank \(0\).
Complex multiplication
The elliptic curves in class 231.a do not have complex multiplication.Modular form 231.2.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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 4 & 8 \\ 2 & 1 & 2 & 4 & 2 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 8 & 4 & 2 & 1 & 8 & 4 \\ 4 & 2 & 4 & 8 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
