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SageMath
E = EllipticCurve("q1")
E.isogeny_class()
Elliptic curves in class 5082.q
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 5082.q1 | 5082r4 | \([1, 1, 1, -27409, 1734437]\) | \(1285429208617/614922\) | \(1089371833242\) | \([2]\) | \(15360\) | \(1.2646\) | |
| 5082.q2 | 5082r3 | \([1, 1, 1, -15309, -723315]\) | \(223980311017/4278582\) | \(7579769006502\) | \([2]\) | \(15360\) | \(1.2646\) | |
| 5082.q3 | 5082r2 | \([1, 1, 1, -1999, 16721]\) | \(498677257/213444\) | \(378129066084\) | \([2, 2]\) | \(7680\) | \(0.91803\) | |
| 5082.q4 | 5082r1 | \([1, 1, 1, 421, 2201]\) | \(4657463/3696\) | \(-6547689456\) | \([4]\) | \(3840\) | \(0.57146\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 5082.q have rank \(1\).
Complex multiplication
The elliptic curves in class 5082.q do not have complex multiplication.Modular form 5082.2.a.q
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.
