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SageMath
E = EllipticCurve("bp1")
E.isogeny_class()
Elliptic curves in class 17640.bp
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 17640.bp1 | 17640bg5 | \([0, 0, 0, -10760547, -13582936514]\) | \(784478485879202/221484375\) | \(38903512485600000000\) | \([2]\) | \(786432\) | \(2.7400\) | |
| 17640.bp2 | 17640bg3 | \([0, 0, 0, -758667, -154412426]\) | \(549871953124/200930625\) | \(17646633263468160000\) | \([2, 2]\) | \(393216\) | \(2.3934\) | |
| 17640.bp3 | 17640bg2 | \([0, 0, 0, -326487, 70061866]\) | \(175293437776/4862025\) | \(106751238260486400\) | \([2, 2]\) | \(196608\) | \(2.0469\) | |
| 17640.bp4 | 17640bg1 | \([0, 0, 0, -324282, 71077489]\) | \(2748251600896/2205\) | \(3025828748880\) | \([2]\) | \(98304\) | \(1.7003\) | \(\Gamma_0(N)\)-optimal |
| 17640.bp5 | 17640bg4 | \([0, 0, 0, 70413, 229536286]\) | \(439608956/259416045\) | \(-22783086494526919680\) | \([2]\) | \(393216\) | \(2.3934\) | |
| 17640.bp6 | 17640bg6 | \([0, 0, 0, 2328333, -1092243026]\) | \(7947184069438/7533176175\) | \(-1323194981047023974400\) | \([2]\) | \(786432\) | \(2.7400\) |
Rank
sage: E.rank()
The elliptic curves in class 17640.bp have rank \(0\).
Complex multiplication
The elliptic curves in class 17640.bp do not have complex multiplication.Modular form 17640.2.a.bp
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 8 & 4 \\ 2 & 1 & 2 & 4 & 4 & 2 \\ 4 & 2 & 1 & 2 & 2 & 4 \\ 8 & 4 & 2 & 1 & 4 & 8 \\ 8 & 4 & 2 & 4 & 1 & 8 \\ 4 & 2 & 4 & 8 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
