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SageMath
E = EllipticCurve("dt1")
E.isogeny_class()
Elliptic curves in class 363090.dt
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
363090.dt1 | 363090dt4 | \([1, 0, 1, -5317898, -4720434652]\) | \(141369383441705190409/6345626621880\) | \(746556626437560120\) | \([2]\) | \(11796480\) | \(2.5063\) | |
363090.dt2 | 363090dt3 | \([1, 0, 1, -1652698, 756855908]\) | \(4243415895694547209/351514682293320\) | \(41355350857126804680\) | \([2]\) | \(11796480\) | \(2.5063\) | |
363090.dt3 | 363090dt2 | \([1, 0, 1, -349298, -65850172]\) | \(40061018056412809/7275103617600\) | \(855908665507022400\) | \([2, 2]\) | \(5898240\) | \(2.1597\) | |
363090.dt4 | 363090dt1 | \([1, 0, 1, 42702, -5952572]\) | \(73197245859191/172623360000\) | \(-20308965680640000\) | \([2]\) | \(2949120\) | \(1.8131\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 363090.dt have rank \(1\).
Complex multiplication
The elliptic curves in class 363090.dt do not have complex multiplication.Modular form 363090.2.a.dt
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.