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SageMath
E = EllipticCurve("pq1")
E.isogeny_class()
Elliptic curves in class 310464pq
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 310464.pq4 | 310464pq1 | \([0, 0, 0, -960204, -362037872]\) | \(4354703137/1617\) | \(36355130695876608\) | \([2]\) | \(3932160\) | \(2.1451\) | \(\Gamma_0(N)\)-optimal |
| 310464.pq3 | 310464pq2 | \([0, 0, 0, -1101324, -248633840]\) | \(6570725617/2614689\) | \(58786246335232475136\) | \([2, 2]\) | \(7864320\) | \(2.4916\) | |
| 310464.pq2 | 310464pq3 | \([0, 0, 0, -8016204, 8560923280]\) | \(2533811507137/58110129\) | \(1306494331817717661696\) | \([2, 2]\) | \(15728640\) | \(2.8382\) | |
| 310464.pq6 | 310464pq4 | \([0, 0, 0, 3555636, -1800332912]\) | \(221115865823/190238433\) | \(-4277144771239187054592\) | \([2]\) | \(15728640\) | \(2.8382\) | |
| 310464.pq1 | 310464pq5 | \([0, 0, 0, -127544844, 554424316432]\) | \(10206027697760497/5557167\) | \(124942197021529079808\) | \([2]\) | \(31457280\) | \(3.1848\) | |
| 310464.pq5 | 310464pq6 | \([0, 0, 0, 874356, 26509185808]\) | \(3288008303/13504609503\) | \(-303625135113384194998272\) | \([2]\) | \(31457280\) | \(3.1848\) |
Rank
sage: E.rank()
The elliptic curves in class 310464pq have rank \(1\).
Complex multiplication
The elliptic curves in class 310464pq do not have complex multiplication.Modular form 310464.2.a.pq
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.
