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SageMath
E = EllipticCurve("fw1")
E.isogeny_class()
Elliptic curves in class 304200.fw
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 304200.fw1 | 304200fw4 | \([0, 0, 0, -316380675, -2166022849250]\) | \(31103978031362/195\) | \(21956961068640000000\) | \([2]\) | \(49545216\) | \(3.3169\) | |
| 304200.fw2 | 304200fw3 | \([0, 0, 0, -27390675, -5427139250]\) | \(20183398562/11567205\) | \(1302464973630656160000000\) | \([2]\) | \(49545216\) | \(3.3169\) | |
| 304200.fw3 | 304200fw2 | \([0, 0, 0, -19785675, -33801394250]\) | \(15214885924/38025\) | \(2140803704192400000000\) | \([2, 2]\) | \(24772608\) | \(2.9703\) | |
| 304200.fw4 | 304200fw1 | \([0, 0, 0, -773175, -928781750]\) | \(-3631696/24375\) | \(-343077516697500000000\) | \([2]\) | \(12386304\) | \(2.6237\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 304200.fw have rank \(0\).
Complex multiplication
The elliptic curves in class 304200.fw do not have complex multiplication.Modular form 304200.2.a.fw
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.
