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SageMath
E = EllipticCurve("bg1")
E.isogeny_class()
Elliptic curves in class 18480.bg
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 18480.bg1 | 18480cd5 | \([0, -1, 0, -48787200, 131177938752]\) | \(3135316978843283198764801/571725\) | \(2341785600\) | \([4]\) | \(491520\) | \(2.5923\) | |
| 18480.bg2 | 18480cd4 | \([0, -1, 0, -3049200, 2050417152]\) | \(765458482133960722801/326869475625\) | \(1338857372160000\) | \([2, 4]\) | \(245760\) | \(2.2457\) | |
| 18480.bg3 | 18480cd6 | \([0, -1, 0, -3034080, 2071742400]\) | \(-754127868744065783521/15825714261328125\) | \(-64822125614400000000\) | \([4]\) | \(491520\) | \(2.5923\) | |
| 18480.bg4 | 18480cd3 | \([0, -1, 0, -407120, -52590720]\) | \(1821931919215868881/761147600816295\) | \(3117660572943544320\) | \([2]\) | \(245760\) | \(2.2457\) | |
| 18480.bg5 | 18480cd2 | \([0, -1, 0, -191520, 31752000]\) | \(189674274234120481/3859869269025\) | \(15810024525926400\) | \([2, 2]\) | \(122880\) | \(1.8991\) | |
| 18480.bg6 | 18480cd1 | \([0, -1, 0, 560, 1480192]\) | \(4733169839/231139696095\) | \(-946748195205120\) | \([2]\) | \(61440\) | \(1.5526\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 18480.bg have rank \(0\).
Complex multiplication
The elliptic curves in class 18480.bg do not have complex multiplication.Modular form 18480.2.a.bg
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 4 & 8 \\ 2 & 1 & 2 & 4 & 2 & 4 \\ 4 & 2 & 1 & 8 & 4 & 8 \\ 8 & 4 & 8 & 1 & 2 & 4 \\ 4 & 2 & 4 & 2 & 1 & 2 \\ 8 & 4 & 8 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
