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SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 25047i
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 25047.c6 | 25047i1 | \([1, -1, 1, 33736, -7001134]\) | \(3288008303/18259263\) | \(-23581253302046847\) | \([2]\) | \(122880\) | \(1.8236\) | \(\Gamma_0(N)\)-optimal |
| 25047.c5 | 25047i2 | \([1, -1, 1, -407309, -90094012]\) | \(5786435182177/627352209\) | \(810205283204893521\) | \([2, 2]\) | \(245760\) | \(2.1702\) | |
| 25047.c4 | 25047i3 | \([1, -1, 1, -1534424, 634866356]\) | \(309368403125137/44372288367\) | \(57305389137211816623\) | \([2]\) | \(491520\) | \(2.5168\) | |
| 25047.c2 | 25047i4 | \([1, -1, 1, -6336914, -6138291112]\) | \(21790813729717297/304746849\) | \(393570794137179681\) | \([2, 2]\) | \(491520\) | \(2.5168\) | |
| 25047.c3 | 25047i5 | \([1, -1, 1, -6157229, -6502907914]\) | \(-19989223566735457/2584262514273\) | \(-3337492260670984821537\) | \([4]\) | \(983040\) | \(2.8634\) | |
| 25047.c1 | 25047i6 | \([1, -1, 1, -101390279, -392929443970]\) | \(89254274298475942657/17457\) | \(22545156334833\) | \([2]\) | \(983040\) | \(2.8634\) |
Rank
sage: E.rank()
The elliptic curves in class 25047i have rank \(0\).
Complex multiplication
The elliptic curves in class 25047i do not have complex multiplication.Modular form 25047.2.a.i
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
