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SageMath
E = EllipticCurve("h1")
E.isogeny_class()
Elliptic curves in class 816h
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
816.b5 | 816h1 | \([0, -1, 0, -544, -4352]\) | \(4354703137/352512\) | \(1443889152\) | \([2]\) | \(384\) | \(0.50050\) | \(\Gamma_0(N)\)-optimal |
816.b4 | 816h2 | \([0, -1, 0, -1824, 25344]\) | \(163936758817/30338064\) | \(124264710144\) | \([2, 2]\) | \(768\) | \(0.84708\) | |
816.b2 | 816h3 | \([0, -1, 0, -27744, 1787904]\) | \(576615941610337/27060804\) | \(110841053184\) | \([2, 4]\) | \(1536\) | \(1.1937\) | |
816.b6 | 816h4 | \([0, -1, 0, 3616, 142848]\) | \(1276229915423/2927177028\) | \(-11989717106688\) | \([2]\) | \(1536\) | \(1.1937\) | |
816.b1 | 816h5 | \([0, -1, 0, -443904, 113984640]\) | \(2361739090258884097/5202\) | \(21307392\) | \([4]\) | \(3072\) | \(1.5402\) | |
816.b3 | 816h6 | \([0, -1, 0, -26304, 1980288]\) | \(-491411892194497/125563633938\) | \(-514308644610048\) | \([4]\) | \(3072\) | \(1.5402\) |
Rank
sage: E.rank()
The elliptic curves in class 816h have rank \(1\).
Complex multiplication
The elliptic curves in class 816h do not have complex multiplication.Modular form 816.2.a.h
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.