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SageMath
E = EllipticCurve("bk1")
E.isogeny_class()
Elliptic curves in class 111090bk
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 111090.bh3 | 111090bk1 | \([1, 0, 1, -187013, 31111136]\) | \(4886171981209/270480\) | \(40040747256720\) | \([2]\) | \(811008\) | \(1.6756\) | \(\Gamma_0(N)\)-optimal |
| 111090.bh2 | 111090bk2 | \([1, 0, 1, -197593, 27391208]\) | \(5763259856089/1143116100\) | \(169222208093712900\) | \([2, 2]\) | \(1622016\) | \(2.0222\) | |
| 111090.bh4 | 111090bk3 | \([1, 0, 1, 410757, 162931588]\) | \(51774168853511/107398242630\) | \(-15898794324769748070\) | \([2]\) | \(3244032\) | \(2.3688\) | |
| 111090.bh1 | 111090bk4 | \([1, 0, 1, -975223, -346182244]\) | \(692895692874169/51420783750\) | \(7612121435508003750\) | \([2]\) | \(3244032\) | \(2.3688\) |
Rank
sage: E.rank()
The elliptic curves in class 111090bk have rank \(1\).
Complex multiplication
The elliptic curves in class 111090bk do not have complex multiplication.Modular form 111090.2.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.
