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SageMath
E = EllipticCurve("ew1")
E.isogeny_class()
Elliptic curves in class 323400.ew
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 323400.ew1 | 323400ew3 | \([0, 1, 0, -8644008, 9778789488]\) | \(18972782339618/396165\) | \(1491469314720000000\) | \([2]\) | \(11796480\) | \(2.6056\) | |
| 323400.ew2 | 323400ew4 | \([0, 1, 0, -2274008, -1175650512]\) | \(345431270018/41507235\) | \(156265110096480000000\) | \([2]\) | \(11796480\) | \(2.6056\) | |
| 323400.ew3 | 323400ew2 | \([0, 1, 0, -559008, 141469488]\) | \(10262905636/1334025\) | \(2511147315600000000\) | \([2, 2]\) | \(5898240\) | \(2.2590\) | |
| 323400.ew4 | 323400ew1 | \([0, 1, 0, 53492, 11619488]\) | \(35969456/144375\) | \(-67942297500000000\) | \([2]\) | \(2949120\) | \(1.9125\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 323400.ew have rank \(1\).
Complex multiplication
The elliptic curves in class 323400.ew do not have complex multiplication.Modular form 323400.2.a.ew
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
