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SageMath
E = EllipticCurve("bh1")
E.isogeny_class()
Elliptic curves in class 54978bh
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
54978.q3 | 54978bh1 | \([1, 0, 1, -405697, 89459444]\) | \(62768149033310713/6915442583808\) | \(813594904542427392\) | \([2]\) | \(1105920\) | \(2.1701\) | \(\Gamma_0(N)\)-optimal |
54978.q2 | 54978bh2 | \([1, 0, 1, -1538577, -638755820]\) | \(3423676911662954233/483711578981136\) | \(56908183555551669264\) | \([2, 2]\) | \(2211840\) | \(2.5166\) | |
54978.q4 | 54978bh3 | \([1, 0, 1, 2509803, -3432138020]\) | \(14861225463775641287/51859390496937804\) | \(-6101205432574235702796\) | \([2]\) | \(4423680\) | \(2.8632\) | |
54978.q1 | 54978bh4 | \([1, 0, 1, -23713037, -44446618996]\) | \(12534210458299016895673/315581882565708\) | \(37127892901972980492\) | \([2]\) | \(4423680\) | \(2.8632\) |
Rank
sage: E.rank()
The elliptic curves in class 54978bh have rank \(1\).
Complex multiplication
The elliptic curves in class 54978bh do not have complex multiplication.Modular form 54978.2.a.bh
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.