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SageMath
E = EllipticCurve("x1")
E.isogeny_class()
Elliptic curves in class 28050x
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
28050.bh5 | 28050x1 | \([1, 0, 1, -7101, 173848]\) | \(2533811507137/625016832\) | \(9765888000000\) | \([2]\) | \(65536\) | \(1.2029\) | \(\Gamma_0(N)\)-optimal |
28050.bh4 | 28050x2 | \([1, 0, 1, -39101, -2834152]\) | \(423108074414017/23284318464\) | \(363817476000000\) | \([2, 2]\) | \(131072\) | \(1.5495\) | |
28050.bh6 | 28050x3 | \([1, 0, 1, 26899, -11414152]\) | \(137763859017023/3683199928848\) | \(-57549998888250000\) | \([2]\) | \(262144\) | \(1.8961\) | |
28050.bh2 | 28050x4 | \([1, 0, 1, -617101, -186638152]\) | \(1663303207415737537/5483698704\) | \(85682792250000\) | \([2, 2]\) | \(262144\) | \(1.8961\) | |
28050.bh3 | 28050x5 | \([1, 0, 1, -608601, -192027152]\) | \(-1595514095015181697/95635786040388\) | \(-1494309156881062500\) | \([2]\) | \(524288\) | \(2.2426\) | |
28050.bh1 | 28050x6 | \([1, 0, 1, -9873601, -11942393152]\) | \(6812873765474836663297/74052\) | \(1157062500\) | \([2]\) | \(524288\) | \(2.2426\) |
Rank
sage: E.rank()
The elliptic curves in class 28050x have rank \(0\).
Complex multiplication
The elliptic curves in class 28050x do not have complex multiplication.Modular form 28050.2.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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.