Show commands:
SageMath
E = EllipticCurve("m1")
E.isogeny_class()
Elliptic curves in class 324870m
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 324870.m7 | 324870m1 | \([1, 1, 0, -3110153, 1508574453]\) | \(28280100765151839241/7994847656250000\) | \(940585831910156250000\) | \([2]\) | \(14598144\) | \(2.7313\) | \(\Gamma_0(N)\)-optimal |
| 324870.m6 | 324870m2 | \([1, 1, 0, -18422653, -29248113047]\) | \(5877491705974396839241/261806444735062500\) | \(30801266416635368062500\) | \([2, 2]\) | \(29196288\) | \(3.0778\) | |
| 324870.m4 | 324870m3 | \([1, 1, 0, -231419528, 1354931051328]\) | \(11650256451486052494789241/580277967360000\) | \(68269122581936640000\) | \([2]\) | \(43794432\) | \(3.2806\) | |
| 324870.m8 | 324870m4 | \([1, 1, 0, 9691097, -110536209797]\) | \(855567391070976980759/45363085180055574750\) | \(-5336921608348358313762750\) | \([2]\) | \(58392576\) | \(3.4244\) | |
| 324870.m2 | 324870m5 | \([1, 1, 0, -291536403, -1916081766297]\) | \(23292378980986805290659241/49479832772574750\) | \(5821252845860646762750\) | \([2]\) | \(58392576\) | \(3.4244\) | |
| 324870.m3 | 324870m6 | \([1, 1, 0, -231811528, 1350110000128]\) | \(11709559667189768059461241/82207646338733697600\) | \(9671647384105680788942400\) | \([2, 2]\) | \(87588864\) | \(3.6272\) | |
| 324870.m5 | 324870m7 | \([1, 1, 0, -87369328, 3013304156248]\) | \(-626920492174472718626041/32979221374608565962360\) | \(-3879972415501323176905691640\) | \([2]\) | \(175177728\) | \(3.9737\) | |
| 324870.m1 | 324870m8 | \([1, 1, 0, -382525728, -621623592792]\) | \(52615951054626272117608441/29030877531795041917560\) | \(3415453710738154886559016440\) | \([2]\) | \(175177728\) | \(3.9737\) |
Rank
sage: E.rank()
The elliptic curves in class 324870m have rank \(1\).
Complex multiplication
The elliptic curves in class 324870m do not have complex multiplication.Modular form 324870.2.a.m
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
