Show commands:
SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 30030f
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 30030.f4 | 30030f1 | \([1, 1, 0, 9028, -231216]\) | \(81362296193000759/70937298432000\) | \(-70937298432000\) | \([2]\) | \(110592\) | \(1.3452\) | \(\Gamma_0(N)\)-optimal |
| 30030.f3 | 30030f2 | \([1, 1, 0, -45052, -2102384]\) | \(10113023132079746761/3976941969000000\) | \(3976941969000000\) | \([2, 2]\) | \(221184\) | \(1.6918\) | |
| 30030.f2 | 30030f3 | \([1, 1, 0, -325332, 69817464]\) | \(3808080733410903748681/89790873046875000\) | \(89790873046875000\) | \([2]\) | \(442368\) | \(2.0384\) | |
| 30030.f1 | 30030f4 | \([1, 1, 0, -630052, -192695384]\) | \(27660114443410429586761/9875086818597000\) | \(9875086818597000\) | \([2]\) | \(442368\) | \(2.0384\) |
Rank
sage: E.rank()
The elliptic curves in class 30030f have rank \(1\).
Complex multiplication
The elliptic curves in class 30030f do not have complex multiplication.Modular form 30030.2.a.f
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
