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SageMath
E = EllipticCurve("w1")
E.isogeny_class()
Elliptic curves in class 4032.w
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 4032.w1 | 4032l6 | \([0, 0, 0, -1572780, 759189296]\) | \(2251439055699625/25088\) | \(4794391461888\) | \([2]\) | \(27648\) | \(2.0021\) | |
| 4032.w2 | 4032l5 | \([0, 0, 0, -98220, 11882288]\) | \(-548347731625/1835008\) | \(-350675489783808\) | \([2]\) | \(13824\) | \(1.6556\) | |
| 4032.w3 | 4032l4 | \([0, 0, 0, -20460, 923312]\) | \(4956477625/941192\) | \(179864592187392\) | \([2]\) | \(9216\) | \(1.4528\) | |
| 4032.w4 | 4032l2 | \([0, 0, 0, -6060, -181456]\) | \(128787625/98\) | \(18728091648\) | \([2]\) | \(3072\) | \(0.90352\) | |
| 4032.w5 | 4032l1 | \([0, 0, 0, -300, -4048]\) | \(-15625/28\) | \(-5350883328\) | \([2]\) | \(1536\) | \(0.55694\) | \(\Gamma_0(N)\)-optimal |
| 4032.w6 | 4032l3 | \([0, 0, 0, 2580, 84656]\) | \(9938375/21952\) | \(-4195092529152\) | \([2]\) | \(4608\) | \(1.1062\) |
Rank
sage: E.rank()
The elliptic curves in class 4032.w have rank \(1\).
Complex multiplication
The elliptic curves in class 4032.w do not have complex multiplication.Modular form 4032.2.a.w
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 3 & 9 & 18 & 6 \\ 2 & 1 & 6 & 18 & 9 & 3 \\ 3 & 6 & 1 & 3 & 6 & 2 \\ 9 & 18 & 3 & 1 & 2 & 6 \\ 18 & 9 & 6 & 2 & 1 & 3 \\ 6 & 3 & 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
