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SageMath
E = EllipticCurve("h1")
E.isogeny_class()
Elliptic curves in class 18240h
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
18240.e4 | 18240h1 | \([0, -1, 0, 597439, -378137535]\) | \(89962967236397039/287450726400000\) | \(-75353483221401600000\) | \([2]\) | \(460800\) | \(2.4973\) | \(\Gamma_0(N)\)-optimal |
18240.e3 | 18240h2 | \([0, -1, 0, -5628481, -4426230719]\) | \(75224183150104868881/11219310000000000\) | \(2941074800640000000000\) | \([2]\) | \(921600\) | \(2.8438\) | |
18240.e2 | 18240h3 | \([0, -1, 0, -211293761, -1182095022015]\) | \(-3979640234041473454886161/1471455901872240\) | \(-385733335940396482560\) | \([2]\) | \(2304000\) | \(3.3020\) | |
18240.e1 | 18240h4 | \([0, -1, 0, -3380700481, -75657448009919]\) | \(16300610738133468173382620881/2228489100\) | \(584185046630400\) | \([2]\) | \(4608000\) | \(3.6485\) |
Rank
sage: E.rank()
The elliptic curves in class 18240h have rank \(0\).
Complex multiplication
The elliptic curves in class 18240h do not have complex multiplication.Modular form 18240.2.a.h
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 5 & 10 \\ 2 & 1 & 10 & 5 \\ 5 & 10 & 1 & 2 \\ 10 & 5 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.