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SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 103428r
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 103428.s2 | 103428r1 | \([0, 0, 0, -4796220, -4042890839]\) | \(216727177216000/2738853\) | \(154197150496738128\) | \([2]\) | \(1935360\) | \(2.4444\) | \(\Gamma_0(N)\)-optimal |
| 103428.s3 | 103428r2 | \([0, 0, 0, -4666935, -4271130578]\) | \(-12479332642000/1526829993\) | \(-1375366019065964460288\) | \([2]\) | \(3870720\) | \(2.7909\) | |
| 103428.s1 | 103428r3 | \([0, 0, 0, -7534020, 1074440653]\) | \(840033089536000/477272151837\) | \(26870374505207815023312\) | \([2]\) | \(5806080\) | \(2.9937\) | |
| 103428.s4 | 103428r4 | \([0, 0, 0, 29829345, 8554586326]\) | \(3258571509326000/1920843121977\) | \(-1730292416336980668287232\) | \([2]\) | \(11612160\) | \(3.3402\) |
Rank
sage: E.rank()
The elliptic curves in class 103428r have rank \(0\).
Complex multiplication
The elliptic curves in class 103428r do not have complex multiplication.Modular form 103428.2.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.
