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SageMath
E = EllipticCurve("er1")
E.isogeny_class()
Elliptic curves in class 129360.er
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
129360.er1 | 129360ha4 | \([0, 1, 0, -450813736, 3684055016564]\) | \(21026497979043461623321/161783881875\) | \(77962084019043840000\) | \([2]\) | \(23592960\) | \(3.4094\) | |
129360.er2 | 129360ha2 | \([0, 1, 0, -28194616, 57475824020]\) | \(5143681768032498601/14238434358225\) | \(6861363461369090150400\) | \([2, 2]\) | \(11796480\) | \(3.0628\) | |
129360.er3 | 129360ha3 | \([0, 1, 0, -17081416, 103257762740]\) | \(-1143792273008057401/8897444448004035\) | \(-4287592209871776619376640\) | \([2]\) | \(23592960\) | \(3.4094\) | |
129360.er4 | 129360ha1 | \([0, 1, 0, -2475496, 101611124]\) | \(3481467828171481/2005331497785\) | \(966349805088388976640\) | \([2]\) | \(5898240\) | \(2.7162\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 129360.er have rank \(1\).
Complex multiplication
The elliptic curves in class 129360.er do not have complex multiplication.Modular form 129360.2.a.er
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.