Show commands:
SageMath
E = EllipticCurve("bf1")
E.isogeny_class()
Elliptic curves in class 20592bf
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 20592.bb4 | 20592bf1 | \([0, 0, 0, 51765, 3259514]\) | \(5137417856375/4510142208\) | \(-13467212470812672\) | \([2]\) | \(110592\) | \(1.7824\) | \(\Gamma_0(N)\)-optimal |
| 20592.bb3 | 20592bf2 | \([0, 0, 0, -259275, 28951418]\) | \(645532578015625/252306960048\) | \(753384545791967232\) | \([2]\) | \(221184\) | \(2.1290\) | |
| 20592.bb2 | 20592bf3 | \([0, 0, 0, -537915, -202774678]\) | \(-5764706497797625/2612665516032\) | \(-7801377428223295488\) | \([2]\) | \(331776\) | \(2.3317\) | |
| 20592.bb1 | 20592bf4 | \([0, 0, 0, -9385275, -11065563286]\) | \(30618029936661765625/3678951124992\) | \(10985289196008112128\) | \([2]\) | \(663552\) | \(2.6783\) |
Rank
sage: E.rank()
The elliptic curves in class 20592bf have rank \(0\).
Complex multiplication
The elliptic curves in class 20592bf do not have complex multiplication.Modular form 20592.2.a.bf
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 Cremona numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
