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SageMath
E = EllipticCurve("cc1")
E.isogeny_class()
Elliptic curves in class 169050cc
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 169050.jf4 | 169050cc1 | \([1, 0, 0, 5689487, -36786584983]\) | \(11079872671250375/324440155855872\) | \(-596407185879491952000000\) | \([2]\) | \(33177600\) | \(3.2422\) | \(\Gamma_0(N)\)-optimal |
| 169050.jf2 | 169050cc2 | \([1, 0, 0, -137194513, -589319012983]\) | \(155355156733986861625/8291568305839392\) | \(15242104993964041084500000\) | \([2]\) | \(66355200\) | \(3.5887\) | |
| 169050.jf3 | 169050cc3 | \([1, 0, 0, -51364888, 1010697507392]\) | \(-8152944444844179625/235342826399858688\) | \(-432622627861202731008000000\) | \([2]\) | \(99532800\) | \(3.7915\) | |
| 169050.jf1 | 169050cc4 | \([1, 0, 0, -1857700888, 30668928291392]\) | \(385693937170561837203625/2159357734550274048\) | \(3969473095501643616768000000\) | \([2]\) | \(199065600\) | \(4.1380\) |
Rank
sage: E.rank()
The elliptic curves in class 169050cc have rank \(0\).
Complex multiplication
The elliptic curves in class 169050cc do not have complex multiplication.Modular form 169050.2.a.cc
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.
