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SageMath
E = EllipticCurve("ie1")
E.isogeny_class()
Elliptic curves in class 277200.ie
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 277200.ie1 | 277200ie4 | \([0, 0, 0, -3666306675, -75677907640750]\) | \(233632133015204766393938/29145526885986328125\) | \(679906851196289062500000000000\) | \([2]\) | \(377487360\) | \(4.4540\) | |
| 277200.ie2 | 277200ie2 | \([0, 0, 0, -915789675, 9442341958250]\) | \(7282213870869695463556/912102595400390625\) | \(10638764672750156250000000000\) | \([2, 2]\) | \(188743680\) | \(4.1074\) | |
| 277200.ie3 | 277200ie1 | \([0, 0, 0, -886265175, 10155151961750]\) | \(26401417552259125806544/507547744790625\) | \(1480009223809462500000000\) | \([2]\) | \(94371840\) | \(3.7608\) | \(\Gamma_0(N)\)-optimal |
| 277200.ie4 | 277200ie3 | \([0, 0, 0, 1362335325, 48942751333250]\) | \(11986661998777424518222/51295853620928503125\) | \(-1196629673269020120900000000000\) | \([2]\) | \(377487360\) | \(4.4540\) |
Rank
sage: E.rank()
The elliptic curves in class 277200.ie have rank \(1\).
Complex multiplication
The elliptic curves in class 277200.ie do not have complex multiplication.Modular form 277200.2.a.ie
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
