Show commands:
SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 19320.d
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 19320.d1 | 19320o3 | \([0, -1, 0, -13616, 615756]\) | \(136324616160098/88149915\) | \(180531025920\) | \([2]\) | \(32768\) | \(1.1002\) | |
| 19320.d2 | 19320o2 | \([0, -1, 0, -1016, 5916]\) | \(113378906596/52490025\) | \(53749785600\) | \([2, 2]\) | \(16384\) | \(0.75359\) | |
| 19320.d3 | 19320o1 | \([0, -1, 0, -516, -4284]\) | \(59466754384/905625\) | \(231840000\) | \([2]\) | \(8192\) | \(0.40702\) | \(\Gamma_0(N)\)-optimal |
| 19320.d4 | 19320o4 | \([0, -1, 0, 3584, 40876]\) | \(2485287189502/1811590515\) | \(-3710137374720\) | \([2]\) | \(32768\) | \(1.1002\) |
Rank
sage: E.rank()
The elliptic curves in class 19320.d have rank \(1\).
Complex multiplication
The elliptic curves in class 19320.d do not have complex multiplication.Modular form 19320.2.a.d
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.
