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SageMath
E = EllipticCurve("t1")
E.isogeny_class()
Elliptic curves in class 58870.t
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
58870.t1 | 58870r4 | \([1, -1, 1, -642682, 198125421]\) | \(49354130009241/99019340\) | \(58899012662028140\) | \([2]\) | \(645120\) | \(2.1051\) | |
58870.t2 | 58870r3 | \([1, -1, 1, -541762, -152564851]\) | \(29563822919961/174072500\) | \(103542382544772500\) | \([2]\) | \(645120\) | \(2.1051\) | |
58870.t3 | 58870r2 | \([1, -1, 1, -53982, 793181]\) | \(29246580441/16483600\) | \(9804829694035600\) | \([2, 2]\) | \(322560\) | \(1.7585\) | |
58870.t4 | 58870r1 | \([1, -1, 1, 13298, 93469]\) | \(437245479/259840\) | \(-154558891728640\) | \([4]\) | \(161280\) | \(1.4120\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 58870.t have rank \(1\).
Complex multiplication
The elliptic curves in class 58870.t do not have complex multiplication.Modular form 58870.2.a.t
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.