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SageMath
E = EllipticCurve("n1")
E.isogeny_class()
Elliptic curves in class 85176.n
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 85176.n1 | 85176bw6 | \([0, 0, 0, -4305136251, 108724761767414]\) | \(1224522642327678150914/66339\) | \(478064521955284992\) | \([2]\) | \(24772608\) | \(3.7831\) | |
| 85176.n2 | 85176bw4 | \([0, 0, 0, -269071491, 1698818100590]\) | \(597914615076708388/4400862921\) | \(15857161160995825542144\) | \([2, 2]\) | \(12386304\) | \(3.4365\) | |
| 85176.n3 | 85176bw5 | \([0, 0, 0, -263535051, 1772072952806]\) | \(-280880296871140514/25701087819771\) | \(-185211990869467747735001088\) | \([2]\) | \(24772608\) | \(3.7831\) | |
| 85176.n4 | 85176bw3 | \([0, 0, 0, -57409131, -137947911946]\) | \(5807363790481348/1079211743883\) | \(3888608860760305831898112\) | \([2]\) | \(12386304\) | \(3.4365\) | |
| 85176.n5 | 85176bw2 | \([0, 0, 0, -17163471, 25393123730]\) | \(620742479063632/49991146569\) | \(45031944984295758338304\) | \([2, 2]\) | \(6193152\) | \(3.0899\) | |
| 85176.n6 | 85176bw1 | \([0, 0, 0, 1096134, 1798062149]\) | \(2587063175168/26304786963\) | \(-1480956880167846205488\) | \([2]\) | \(3096576\) | \(2.7433\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 85176.n have rank \(1\).
Complex multiplication
The elliptic curves in class 85176.n do not have complex multiplication.Modular form 85176.2.a.n
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 LMFDB numbering.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 4 & 8 \\ 2 & 1 & 2 & 4 & 2 & 4 \\ 4 & 2 & 1 & 8 & 4 & 8 \\ 8 & 4 & 8 & 1 & 2 & 4 \\ 4 & 2 & 4 & 2 & 1 & 2 \\ 8 & 4 & 8 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
