Show commands:
SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 388311d
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 388311.d4 | 388311d1 | \([1, 0, 0, -57189, 5257560]\) | \(4354703137/1617\) | \(7680918557697\) | \([2]\) | \(1331200\) | \(1.4399\) | \(\Gamma_0(N)\)-optimal |
| 388311.d3 | 388311d2 | \([1, 0, 0, -65594, 3608499]\) | \(6570725617/2614689\) | \(12420045307796049\) | \([2, 2]\) | \(2662400\) | \(1.7864\) | |
| 388311.d6 | 388311d3 | \([1, 0, 0, 211771, 26186010]\) | \(221115865823/190238433\) | \(-903652387394494353\) | \([2]\) | \(5324800\) | \(2.1330\) | |
| 388311.d2 | 388311d4 | \([1, 0, 0, -477439, -124475296]\) | \(2533811507137/58110129\) | \(276029170207957089\) | \([2, 2]\) | \(5324800\) | \(2.1330\) | |
| 388311.d5 | 388311d5 | \([1, 0, 0, 52076, -385314385]\) | \(3288008303/13504609503\) | \(-64148302873249202223\) | \([2]\) | \(10649600\) | \(2.4796\) | |
| 388311.d1 | 388311d6 | \([1, 0, 0, -7596474, -8059351707]\) | \(10206027697760497/5557167\) | \(26397122534645247\) | \([2]\) | \(10649600\) | \(2.4796\) |
Rank
sage: E.rank()
The elliptic curves in class 388311d have rank \(1\).
Complex multiplication
The elliptic curves in class 388311d do not have complex multiplication.Modular form 388311.2.a.d
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
