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SageMath
E = EllipticCurve("x1")
E.isogeny_class()
Elliptic curves in class 19074.x
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
19074.x1 | 19074p3 | \([1, 1, 1, -2908791, -1910699799]\) | \(112763292123580561/1932612\) | \(46648555500228\) | \([2]\) | \(512000\) | \(2.1645\) | |
19074.x2 | 19074p4 | \([1, 1, 1, -2905901, -1914682219]\) | \(-112427521449300721/466873642818\) | \(-11269194767800829442\) | \([2]\) | \(1024000\) | \(2.5110\) | |
19074.x3 | 19074p1 | \([1, 1, 1, -13011, 410961]\) | \(10091699281/2737152\) | \(66068195263488\) | \([2]\) | \(102400\) | \(1.3597\) | \(\Gamma_0(N)\)-optimal |
19074.x4 | 19074p2 | \([1, 1, 1, 33229, 2722961]\) | \(168105213359/228637728\) | \(-5518758935603232\) | \([2]\) | \(204800\) | \(1.7063\) |
Rank
sage: E.rank()
The elliptic curves in class 19074.x have rank \(0\).
Complex multiplication
The elliptic curves in class 19074.x do not have complex multiplication.Modular form 19074.2.a.x
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 5 & 10 \\ 2 & 1 & 10 & 5 \\ 5 & 10 & 1 & 2 \\ 10 & 5 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.