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SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 18975.r
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 18975.r1 | 18975e5 | \([1, 1, 0, -2327600, -1367788125]\) | \(89254274298475942657/17457\) | \(272765625\) | \([2]\) | \(131072\) | \(1.9198\) | |
| 18975.r2 | 18975e3 | \([1, 1, 0, -145475, -21417000]\) | \(21790813729717297/304746849\) | \(4761669515625\) | \([2, 2]\) | \(65536\) | \(1.5733\) | |
| 18975.r3 | 18975e6 | \([1, 1, 0, -141350, -22683375]\) | \(-19989223566735457/2584262514273\) | \(-40379101785515625\) | \([2]\) | \(131072\) | \(1.9198\) | |
| 18975.r4 | 18975e4 | \([1, 1, 0, -35225, 2192250]\) | \(309368403125137/44372288367\) | \(693317005734375\) | \([2]\) | \(65536\) | \(1.5733\) | |
| 18975.r5 | 18975e2 | \([1, 1, 0, -9350, -317625]\) | \(5786435182177/627352209\) | \(9802378265625\) | \([2, 2]\) | \(32768\) | \(1.2267\) | |
| 18975.r6 | 18975e1 | \([1, 1, 0, 775, -24000]\) | \(3288008303/18259263\) | \(-285300984375\) | \([2]\) | \(16384\) | \(0.88011\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 18975.r have rank \(1\).
Complex multiplication
The elliptic curves in class 18975.r do not have complex multiplication.Modular form 18975.2.a.r
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 4 & 8 \\ 2 & 1 & 2 & 4 & 2 & 4 \\ 4 & 2 & 1 & 8 & 4 & 8 \\ 8 & 4 & 8 & 1 & 2 & 4 \\ 4 & 2 & 4 & 2 & 1 & 2 \\ 8 & 4 & 8 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
