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SageMath
E = EllipticCurve("bi1")
E.isogeny_class()
Elliptic curves in class 47190.bi
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 47190.bi1 | 47190bg6 | \([1, 0, 1, -719804803, 7433044259606]\) | \(23281546263261052473907201/1115400\) | \(1975999139400\) | \([2]\) | \(7372800\) | \(3.2499\) | |
| 47190.bi2 | 47190bg4 | \([1, 0, 1, -44987803, 116138492006]\) | \(5683972151443376419201/1244117160000\) | \(2204029440086760000\) | \([2, 2]\) | \(3686400\) | \(2.9033\) | |
| 47190.bi3 | 47190bg5 | \([1, 0, 1, -44828083, 117004110518]\) | \(-5623647484692626737921/84122230603125000\) | \(-149027662969502728125000\) | \([2]\) | \(7372800\) | \(3.2499\) | |
| 47190.bi4 | 47190bg2 | \([1, 0, 1, -2821723, 1800949478]\) | \(1402524686897642881/20523074457600\) | \(36357878309180313600\) | \([2, 2]\) | \(1843200\) | \(2.5568\) | |
| 47190.bi5 | 47190bg1 | \([1, 0, 1, -343643, -33820954]\) | \(2533309721804161/1187575234560\) | \(2103861970112348160\) | \([2]\) | \(921600\) | \(2.2102\) | \(\Gamma_0(N)\)-optimal |
| 47190.bi6 | 47190bg3 | \([1, 0, 1, -304923, 4898626918]\) | \(-1769848555063681/5850659851882560\) | \(-10364800817860919876160\) | \([2]\) | \(3686400\) | \(2.9033\) |
Rank
sage: E.rank()
The elliptic curves in class 47190.bi have rank \(0\).
Complex multiplication
The elliptic curves in class 47190.bi do not have complex multiplication.Modular form 47190.2.a.bi
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
