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SageMath
E = EllipticCurve("lp1")
E.isogeny_class()
Elliptic curves in class 383040lp
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 383040.lp4 | 383040lp1 | \([0, 0, 0, -5820492, 5404894864]\) | \(114113060120923921/124104960\) | \(23716827192360960\) | \([2]\) | \(11796480\) | \(2.4295\) | \(\Gamma_0(N)\)-optimal |
| 383040.lp3 | 383040lp2 | \([0, 0, 0, -5866572, 5314965136]\) | \(116844823575501841/3760263939600\) | \(718597629403044249600\) | \([2, 2]\) | \(23592960\) | \(2.7761\) | |
| 383040.lp5 | 383040lp3 | \([0, 0, 0, 1794228, 18206559376]\) | \(3342636501165359/751262567039460\) | \(-143568512318640315432960\) | \([2]\) | \(47185920\) | \(3.1226\) | |
| 383040.lp2 | 383040lp4 | \([0, 0, 0, -14264652, -13332131696]\) | \(1679731262160129361/570261564022500\) | \(108978681983114280960000\) | \([2, 2]\) | \(47185920\) | \(3.1226\) | |
| 383040.lp6 | 383040lp5 | \([0, 0, 0, 41878068, -92403538544]\) | \(42502666283088696719/43898058864843750\) | \(-8389049689694822400000000\) | \([4]\) | \(94371840\) | \(3.4692\) | |
| 383040.lp1 | 383040lp6 | \([0, 0, 0, -204776652, -1127674922096]\) | \(4969327007303723277361/1123462695162150\) | \(214697064470467667558400\) | \([2]\) | \(94371840\) | \(3.4692\) |
Rank
sage: E.rank()
The elliptic curves in class 383040lp have rank \(1\).
Complex multiplication
The elliptic curves in class 383040lp do not have complex multiplication.Modular form 383040.2.a.lp
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
