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SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 133952r
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 133952.bm3 | 133952r1 | \([0, 0, 0, -36284, -2660240]\) | \(322440248841552/27209\) | \(445792256\) | \([2]\) | \(143360\) | \(1.1017\) | \(\Gamma_0(N)\)-optimal |
| 133952.bm2 | 133952r2 | \([0, 0, 0, -36364, -2647920]\) | \(81144432781668/740329681\) | \(48518245974016\) | \([2, 2]\) | \(286720\) | \(1.4482\) | |
| 133952.bm1 | 133952r3 | \([0, 0, 0, -63404, 1819088]\) | \(215062038362754/113550802729\) | \(14883330815295488\) | \([4]\) | \(573440\) | \(1.7948\) | |
| 133952.bm4 | 133952r4 | \([0, 0, 0, -10604, -6326448]\) | \(-1006057824354/131332646081\) | \(-17214032587128832\) | \([2]\) | \(573440\) | \(1.7948\) |
Rank
sage: E.rank()
The elliptic curves in class 133952r have rank \(1\).
Complex multiplication
The elliptic curves in class 133952r do not have complex multiplication.Modular form 133952.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
