Show commands:
SageMath
E = EllipticCurve("z1")
E.isogeny_class()
Elliptic curves in class 4032z
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 4032.r5 | 4032z1 | \([0, 0, 0, -300, 4048]\) | \(-15625/28\) | \(-5350883328\) | \([2]\) | \(1536\) | \(0.55694\) | \(\Gamma_0(N)\)-optimal |
| 4032.r4 | 4032z2 | \([0, 0, 0, -6060, 181456]\) | \(128787625/98\) | \(18728091648\) | \([2]\) | \(3072\) | \(0.90352\) | |
| 4032.r6 | 4032z3 | \([0, 0, 0, 2580, -84656]\) | \(9938375/21952\) | \(-4195092529152\) | \([2]\) | \(4608\) | \(1.1062\) | |
| 4032.r3 | 4032z4 | \([0, 0, 0, -20460, -923312]\) | \(4956477625/941192\) | \(179864592187392\) | \([2]\) | \(9216\) | \(1.4528\) | |
| 4032.r2 | 4032z5 | \([0, 0, 0, -98220, -11882288]\) | \(-548347731625/1835008\) | \(-350675489783808\) | \([2]\) | \(13824\) | \(1.6556\) | |
| 4032.r1 | 4032z6 | \([0, 0, 0, -1572780, -759189296]\) | \(2251439055699625/25088\) | \(4794391461888\) | \([2]\) | \(27648\) | \(2.0021\) |
Rank
sage: E.rank()
The elliptic curves in class 4032z have rank \(1\).
Complex multiplication
The elliptic curves in class 4032z do not have complex multiplication.Modular form 4032.2.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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 3 & 6 & 9 & 18 \\ 2 & 1 & 6 & 3 & 18 & 9 \\ 3 & 6 & 1 & 2 & 3 & 6 \\ 6 & 3 & 2 & 1 & 6 & 3 \\ 9 & 18 & 3 & 6 & 1 & 2 \\ 18 & 9 & 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
