Show commands:
SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 122034d
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
122034.d3 | 122034d1 | \([1, 1, 0, -83243, -6772995]\) | \(10091699281/2737152\) | \(17302531512296448\) | \([2]\) | \(1612800\) | \(1.8237\) | \(\Gamma_0(N)\)-optimal |
122034.d4 | 122034d2 | \([1, 1, 0, 212597, -43752995]\) | \(168105213359/228637728\) | \(-1445302085386512672\) | \([2]\) | \(3225600\) | \(2.1703\) | |
122034.d1 | 122034d3 | \([1, 1, 0, -18610223, 30893456145]\) | \(112763292123580561/1932612\) | \(12216742084853988\) | \([2]\) | \(8064000\) | \(2.6285\) | |
122034.d2 | 122034d4 | \([1, 1, 0, -18591733, 30957930775]\) | \(-112427521449300721/466873642818\) | \(-2951277794261729432082\) | \([2]\) | \(16128000\) | \(2.9750\) |
Rank
sage: E.rank()
The elliptic curves in class 122034d have rank \(1\).
Complex multiplication
The elliptic curves in class 122034d do not have complex multiplication.Modular form 122034.2.a.d
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 5 & 10 \\ 2 & 1 & 10 & 5 \\ 5 & 10 & 1 & 2 \\ 10 & 5 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.