Show commands:
SageMath
E = EllipticCurve("bl1")
E.isogeny_class()
Elliptic curves in class 6762.bl
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 6762.bl1 | 6762bh5 | \([1, 0, 0, -3877567, 2938562717]\) | \(54804145548726848737/637608031452\) | \(75013947292296348\) | \([4]\) | \(196608\) | \(2.3889\) | |
| 6762.bl2 | 6762bh3 | \([1, 0, 0, -867987, -311320143]\) | \(614716917569296417/19093020912\) | \(2246274817275888\) | \([2]\) | \(98304\) | \(2.0424\) | |
| 6762.bl3 | 6762bh4 | \([1, 0, 0, -248627, 43394385]\) | \(14447092394873377/1439452851984\) | \(169350188583065616\) | \([2, 2]\) | \(98304\) | \(2.0424\) | |
| 6762.bl4 | 6762bh2 | \([1, 0, 0, -56547, -4433535]\) | \(169967019783457/26337394944\) | \(3098568177766656\) | \([2, 2]\) | \(49152\) | \(1.6958\) | |
| 6762.bl5 | 6762bh1 | \([1, 0, 0, 6173, -381823]\) | \(221115865823/664731648\) | \(-78205013655552\) | \([2]\) | \(24576\) | \(1.3492\) | \(\Gamma_0(N)\)-optimal |
| 6762.bl6 | 6762bh6 | \([1, 0, 0, 307033, 209981253]\) | \(27207619911317663/177609314617308\) | \(-20895558255411668892\) | \([2]\) | \(196608\) | \(2.3889\) |
Rank
sage: E.rank()
The elliptic curves in class 6762.bl have rank \(0\).
Complex multiplication
The elliptic curves in class 6762.bl do not have complex multiplication.Modular form 6762.2.a.bl
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 2 & 4 & 8 & 4 \\ 8 & 1 & 4 & 2 & 4 & 8 \\ 2 & 4 & 1 & 2 & 4 & 2 \\ 4 & 2 & 2 & 1 & 2 & 4 \\ 8 & 4 & 4 & 2 & 1 & 8 \\ 4 & 8 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
