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SageMath
E = EllipticCurve("g1")
E.isogeny_class()
Elliptic curves in class 169050.g
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
169050.g1 | 169050ib3 | \([1, 1, 0, -5501500, 4964329000]\) | \(10017490085065009/235066440\) | \(432114556243125000\) | \([2]\) | \(7077888\) | \(2.4960\) | |
169050.g2 | 169050ib4 | \([1, 1, 0, -1483500, -623925000]\) | \(196416765680689/22365315000\) | \(41113389756796875000\) | \([2]\) | \(7077888\) | \(2.4960\) | |
169050.g3 | 169050ib2 | \([1, 1, 0, -356500, 71434000]\) | \(2725812332209/373262400\) | \(686155439025000000\) | \([2, 2]\) | \(3538944\) | \(2.1494\) | |
169050.g4 | 169050ib1 | \([1, 1, 0, 35500, 5970000]\) | \(2691419471/9891840\) | \(-18183829440000000\) | \([2]\) | \(1769472\) | \(1.8028\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 169050.g have rank \(1\).
Complex multiplication
The elliptic curves in class 169050.g do not have complex multiplication.Modular form 169050.2.a.g
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 LMFDB numbering.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.