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SageMath
E = EllipticCurve("x1")
E.isogeny_class()
Elliptic curves in class 38976x
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
38976.bw5 | 38976x1 | \([0, 1, 0, -50177, 4615263]\) | \(-53297461115137/4513839183\) | \(-1183275858788352\) | \([2]\) | \(196608\) | \(1.6372\) | \(\Gamma_0(N)\)-optimal |
38976.bw4 | 38976x2 | \([0, 1, 0, -818497, 284744735]\) | \(231331938231569617/1472026689\) | \(385882964361216\) | \([2, 2]\) | \(393216\) | \(1.9838\) | |
38976.bw3 | 38976x3 | \([0, 1, 0, -834177, 273251295]\) | \(244883173420511137/18418027974129\) | \(4828175525250072576\) | \([2, 2]\) | \(786432\) | \(2.3304\) | |
38976.bw1 | 38976x4 | \([0, 1, 0, -13095937, 18236817503]\) | \(947531277805646290177/38367\) | \(10057678848\) | \([2]\) | \(786432\) | \(2.3304\) | |
38976.bw6 | 38976x5 | \([0, 1, 0, 798783, 1214162847]\) | \(215015459663151503/2552757445339983\) | \(-669190047751204503552\) | \([2]\) | \(1572864\) | \(2.6770\) | |
38976.bw2 | 38976x6 | \([0, 1, 0, -2718017, -1402989537]\) | \(8471112631466271697/1662662681263647\) | \(435857045917177479168\) | \([2]\) | \(1572864\) | \(2.6770\) |
Rank
sage: E.rank()
The elliptic curves in class 38976x have rank \(1\).
Complex multiplication
The elliptic curves in class 38976x do not have complex multiplication.Modular form 38976.2.a.x
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.