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SageMath
E = EllipticCurve("dh1")
E.isogeny_class()
Elliptic curves in class 37200dh
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
37200.ea3 | 37200dh1 | \([0, 1, 0, -97408, -11732812]\) | \(1597099875769/186000\) | \(11904000000000\) | \([2]\) | \(165888\) | \(1.5353\) | \(\Gamma_0(N)\)-optimal |
37200.ea2 | 37200dh2 | \([0, 1, 0, -105408, -9700812]\) | \(2023804595449/540562500\) | \(34596000000000000\) | \([2, 2]\) | \(331776\) | \(1.8819\) | |
37200.ea4 | 37200dh3 | \([0, 1, 0, 266592, -62524812]\) | \(32740359775271/45410156250\) | \(-2906250000000000000\) | \([2]\) | \(663552\) | \(2.2285\) | |
37200.ea1 | 37200dh4 | \([0, 1, 0, -605408, 173299188]\) | \(383432500775449/18701300250\) | \(1196883216000000000\) | \([4]\) | \(663552\) | \(2.2285\) |
Rank
sage: E.rank()
The elliptic curves in class 37200dh have rank \(0\).
Complex multiplication
The elliptic curves in class 37200dh do not have complex multiplication.Modular form 37200.2.a.dh
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.