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SageMath
E = EllipticCurve("bk1")
E.isogeny_class()
Elliptic curves in class 99450bk
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 99450.bw3 | 99450bk1 | \([1, -1, 0, -46917, -3899259]\) | \(1002702430729/159120\) | \(1812476250000\) | \([2]\) | \(393216\) | \(1.3625\) | \(\Gamma_0(N)\)-optimal |
| 99450.bw2 | 99450bk2 | \([1, -1, 0, -51417, -3102759]\) | \(1319778683209/395612100\) | \(4506269076562500\) | \([2, 2]\) | \(786432\) | \(1.7090\) | |
| 99450.bw4 | 99450bk3 | \([1, -1, 0, 139833, -20889009]\) | \(26546265663191/31856082570\) | \(-362860690523906250\) | \([2]\) | \(1572864\) | \(2.0556\) | |
| 99450.bw1 | 99450bk4 | \([1, -1, 0, -314667, 65605491]\) | \(302503589987689/12214946250\) | \(139135872128906250\) | \([2]\) | \(1572864\) | \(2.0556\) |
Rank
sage: E.rank()
The elliptic curves in class 99450bk have rank \(0\).
Complex multiplication
The elliptic curves in class 99450bk do not have complex multiplication.Modular form 99450.2.a.bk
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
