Show commands:
SageMath
E = EllipticCurve("z1")
E.isogeny_class()
Elliptic curves in class 58482z
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 58482.v4 | 58482z1 | \([1, -1, 1, 1015, 1675]\) | \(3375/2\) | \(-68592894498\) | \([]\) | \(43092\) | \(0.76863\) | \(\Gamma_0(N)\)-optimal |
| 58482.v3 | 58482z2 | \([1, -1, 1, -15230, 761941]\) | \(-140625/8\) | \(-22224097817352\) | \([]\) | \(129276\) | \(1.3179\) | |
| 58482.v1 | 58482z3 | \([1, -1, 1, -388865, -93238127]\) | \(-189613868625/128\) | \(-4389945247872\) | \([]\) | \(301644\) | \(1.7416\) | |
| 58482.v2 | 58482z4 | \([1, -1, 1, -307640, -133337285]\) | \(-1159088625/2097152\) | \(-5825913898231922688\) | \([]\) | \(904932\) | \(2.2909\) |
Rank
sage: E.rank()
The elliptic curves in class 58482z have rank \(0\).
Complex multiplication
The elliptic curves in class 58482z do not have complex multiplication.Modular form 58482.2.a.z
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.
\(\left(\begin{array}{rrrr} 1 & 3 & 7 & 21 \\ 3 & 1 & 21 & 7 \\ 7 & 21 & 1 & 3 \\ 21 & 7 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
