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SageMath
E = EllipticCurve("bt1")
E.isogeny_class()
Elliptic curves in class 37485bt
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 37485.bh5 | 37485bt1 | \([1, -1, 0, 15426, 1821343]\) | \(4733169839/19518975\) | \(-1674066771645975\) | \([2]\) | \(196608\) | \(1.6038\) | \(\Gamma_0(N)\)-optimal |
| 37485.bh4 | 37485bt2 | \([1, -1, 0, -163179, 22432360]\) | \(5602762882081/716900625\) | \(61485785748725625\) | \([2, 2]\) | \(393216\) | \(1.9504\) | |
| 37485.bh3 | 37485bt3 | \([1, -1, 0, -659304, -182864165]\) | \(369543396484081/45120132225\) | \(3869778719945349225\) | \([2, 2]\) | \(786432\) | \(2.2969\) | |
| 37485.bh2 | 37485bt4 | \([1, -1, 0, -2524734, 1544690713]\) | \(20751759537944401/418359375\) | \(35881060777734375\) | \([2]\) | \(786432\) | \(2.2969\) | |
| 37485.bh6 | 37485bt5 | \([1, -1, 0, 961371, -941664200]\) | \(1145725929069119/5127181719135\) | \(-439738487712320425335\) | \([2]\) | \(1572864\) | \(2.6435\) | |
| 37485.bh1 | 37485bt6 | \([1, -1, 0, -10217979, -12568995230]\) | \(1375634265228629281/24990412335\) | \(2143330728163502535\) | \([2]\) | \(1572864\) | \(2.6435\) |
Rank
sage: E.rank()
The elliptic curves in class 37485bt have rank \(1\).
Complex multiplication
The elliptic curves in class 37485bt do not have complex multiplication.Modular form 37485.2.a.bt
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.
