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SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 19074j
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
19074.g3 | 19074j1 | \([1, 0, 1, -2392782, -1282066304]\) | \(62768149033310713/6915442583808\) | \(166921972532203882752\) | \([2]\) | \(1105920\) | \(2.6137\) | \(\Gamma_0(N)\)-optimal |
19074.g2 | 19074j2 | \([1, 0, 1, -9074462, 9146699840]\) | \(3423676911662954233/483711578981136\) | \(11675621613756119898384\) | \([2, 2]\) | \(2211840\) | \(2.9603\) | |
19074.g1 | 19074j3 | \([1, 0, 1, -139858522, 636596306096]\) | \(12534210458299016895673/315581882565708\) | \(7617379465579673883852\) | \([4]\) | \(4423680\) | \(3.3069\) | |
19074.g4 | 19074j4 | \([1, 0, 1, 14802718, 49164853520]\) | \(14861225463775641287/51859390496937804\) | \(-1251759616417780532758476\) | \([2]\) | \(4423680\) | \(3.3069\) |
Rank
sage: E.rank()
The elliptic curves in class 19074j have rank \(0\).
Complex multiplication
The elliptic curves in class 19074j do not have complex multiplication.Modular form 19074.2.a.j
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.