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SageMath
E = EllipticCurve("q1")
E.isogeny_class()
Elliptic curves in class 6864q
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 6864.j4 | 6864q1 | \([0, -1, 0, 5752, -122640]\) | \(5137417856375/4510142208\) | \(-18473542483968\) | \([2]\) | \(13824\) | \(1.2331\) | \(\Gamma_0(N)\)-optimal |
| 6864.j3 | 6864q2 | \([0, -1, 0, -28808, -1062672]\) | \(645532578015625/252306960048\) | \(1033449308356608\) | \([2]\) | \(27648\) | \(1.5797\) | |
| 6864.j2 | 6864q3 | \([0, -1, 0, -59768, 7530096]\) | \(-5764706497797625/2612665516032\) | \(-10701477953667072\) | \([2]\) | \(41472\) | \(1.7824\) | |
| 6864.j1 | 6864q4 | \([0, -1, 0, -1042808, 410183280]\) | \(30618029936661765625/3678951124992\) | \(15068983807967232\) | \([2]\) | \(82944\) | \(2.1290\) |
Rank
sage: E.rank()
The elliptic curves in class 6864q have rank \(0\).
Complex multiplication
The elliptic curves in class 6864q do not have complex multiplication.Modular form 6864.2.a.q
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.
