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SageMath
E = EllipticCurve("lj1")
E.isogeny_class()
Elliptic curves in class 381150.lj
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
381150.lj1 | 381150lj3 | \([1, -1, 1, -2504823080, 48252441649547]\) | \(86129359107301290313/9166294368\) | \(184968368292142165500000\) | \([2]\) | \(235929600\) | \(3.8923\) | |
381150.lj2 | 381150lj2 | \([1, -1, 1, -156939080, 750052561547]\) | \(21184262604460873/216872764416\) | \(4376316071746046736000000\) | \([2, 2]\) | \(117964800\) | \(3.5457\) | |
381150.lj3 | 381150lj4 | \([1, -1, 1, -39327080, 1848078193547]\) | \(-333345918055753/72923718045024\) | \(-1471541344305246809941500000\) | \([2]\) | \(235929600\) | \(3.8923\) | |
381150.lj4 | 381150lj1 | \([1, -1, 1, -17547080, -9355054453]\) | \(29609739866953/15259926528\) | \(307932911253209088000000\) | \([2]\) | \(58982400\) | \(3.1991\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 381150.lj have rank \(0\).
Complex multiplication
The elliptic curves in class 381150.lj do not have complex multiplication.Modular form 381150.2.a.lj
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.