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SageMath
E = EllipticCurve("fw1")
E.isogeny_class()
Elliptic curves in class 47040.fw
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 47040.fw1 | 47040hf6 | \([0, 1, 0, -2054145, -1133828865]\) | \(62161150998242/1607445\) | \(24787589110824960\) | \([2]\) | \(786432\) | \(2.2524\) | |
| 47040.fw2 | 47040hf4 | \([0, 1, 0, -133345, -16307425]\) | \(34008619684/4862025\) | \(37487403284889600\) | \([2, 2]\) | \(393216\) | \(1.9059\) | |
| 47040.fw3 | 47040hf2 | \([0, 1, 0, -35345, 2292975]\) | \(2533446736/275625\) | \(531284060160000\) | \([2, 2]\) | \(196608\) | \(1.5593\) | |
| 47040.fw4 | 47040hf1 | \([0, 1, 0, -34365, 2440563]\) | \(37256083456/525\) | \(63248102400\) | \([2]\) | \(98304\) | \(1.2127\) | \(\Gamma_0(N)\)-optimal |
| 47040.fw5 | 47040hf3 | \([0, 1, 0, 46975, 11463423]\) | \(1486779836/8203125\) | \(-63248102400000000\) | \([2]\) | \(393216\) | \(1.9059\) | |
| 47040.fw6 | 47040hf5 | \([0, 1, 0, 219455, -87643585]\) | \(75798394558/259416045\) | \(-4000322457200885760\) | \([2]\) | \(786432\) | \(2.2524\) |
Rank
sage: E.rank()
The elliptic curves in class 47040.fw have rank \(1\).
Complex multiplication
The elliptic curves in class 47040.fw do not have complex multiplication.Modular form 47040.2.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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 8 & 4 \\ 2 & 1 & 2 & 4 & 4 & 2 \\ 4 & 2 & 1 & 2 & 2 & 4 \\ 8 & 4 & 2 & 1 & 4 & 8 \\ 8 & 4 & 2 & 4 & 1 & 8 \\ 4 & 2 & 4 & 8 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
