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SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 37030r
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
37030.p4 | 37030r1 | \([1, 0, 0, -973900, -368430000]\) | \(690080604747409/3406760000\) | \(504322745209640000\) | \([2]\) | \(1317888\) | \(2.2445\) | \(\Gamma_0(N)\)-optimal |
37030.p3 | 37030r2 | \([1, 0, 0, -1502900, 75824200]\) | \(2535986675931409/1450751712200\) | \(214763319433799145800\) | \([2]\) | \(2635776\) | \(2.5911\) | |
37030.p2 | 37030r3 | \([1, 0, 0, -5597360, 4830723772]\) | \(131010595463836369/7704101562500\) | \(1140483523750976562500\) | \([2]\) | \(3953664\) | \(2.7938\) | |
37030.p1 | 37030r4 | \([1, 0, 0, -88253610, 319106317522]\) | \(513516182162686336369/1944885031250\) | \(287912784603886531250\) | \([2]\) | \(7907328\) | \(3.1404\) |
Rank
sage: E.rank()
The elliptic curves in class 37030r have rank \(0\).
Complex multiplication
The elliptic curves in class 37030r do not have complex multiplication.Modular form 37030.2.a.r
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.