Show commands:
SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 31824.k
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 31824.k1 | 31824q4 | \([0, 0, 0, -264891, -52473926]\) | \(2753580869496292/39328497\) | \(29358565696512\) | \([2]\) | \(163840\) | \(1.7237\) | |
| 31824.k2 | 31824q2 | \([0, 0, 0, -17031, -770330]\) | \(2927363579728/320445801\) | \(59802877165824\) | \([2, 2]\) | \(81920\) | \(1.3771\) | |
| 31824.k3 | 31824q1 | \([0, 0, 0, -4026, 85399]\) | \(618724784128/87947613\) | \(1025820958032\) | \([2]\) | \(40960\) | \(1.0305\) | \(\Gamma_0(N)\)-optimal |
| 31824.k4 | 31824q3 | \([0, 0, 0, 22749, -3833390]\) | \(1744147297148/9513325341\) | \(-7101659313755136\) | \([2]\) | \(163840\) | \(1.7237\) |
Rank
sage: E.rank()
The elliptic curves in class 31824.k have rank \(1\).
Complex multiplication
The elliptic curves in class 31824.k do not have complex multiplication.Modular form 31824.2.a.k
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.
