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SageMath
E = EllipticCurve("hx1")
E.isogeny_class()
Elliptic curves in class 383040.hx
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
383040.hx1 | 383040hx3 | \([0, 0, 0, -7225348332, 236393917230256]\) | \(218289391029690300712901881/306514992000\) | \(58575927159816192000\) | \([4]\) | \(165150720\) | \(3.9580\) | |
383040.hx2 | 383040hx4 | \([0, 0, 0, -472600812, 3330985438384]\) | \(61085713691774408830201/10268551781250000000\) | \(1962350804606976000000000000\) | \([2]\) | \(165150720\) | \(3.9580\) | |
383040.hx3 | 383040hx2 | \([0, 0, 0, -451588332, 3693585198256]\) | \(53294746224000958661881/1997017344000000\) | \(381635957562015744000000\) | \([2, 2]\) | \(82575360\) | \(3.6114\) | |
383040.hx4 | 383040hx1 | \([0, 0, 0, -26915052, 63308131504]\) | \(-11283450590382195961/2530373271552000\) | \(-483561862584443338752000\) | \([2]\) | \(41287680\) | \(3.2648\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 383040.hx have rank \(0\).
Complex multiplication
The elliptic curves in class 383040.hx do not have complex multiplication.Modular form 383040.2.a.hx
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.