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SageMath
E = EllipticCurve("z1")
E.isogeny_class()
Elliptic curves in class 3120z
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 3120.w6 | 3120z1 | \([0, 1, 0, 240, -492]\) | \(371694959/249600\) | \(-1022361600\) | \([2]\) | \(1536\) | \(0.41750\) | \(\Gamma_0(N)\)-optimal |
| 3120.w5 | 3120z2 | \([0, 1, 0, -1040, -5100]\) | \(30400540561/15210000\) | \(62300160000\) | \([2, 2]\) | \(3072\) | \(0.76407\) | |
| 3120.w2 | 3120z3 | \([0, 1, 0, -13520, -609132]\) | \(66730743078481/60937500\) | \(249600000000\) | \([2]\) | \(6144\) | \(1.1106\) | |
| 3120.w3 | 3120z4 | \([0, 1, 0, -9040, 324500]\) | \(19948814692561/231344100\) | \(947585433600\) | \([2, 4]\) | \(6144\) | \(1.1106\) | |
| 3120.w1 | 3120z5 | \([0, 1, 0, -144240, 21037140]\) | \(81025909800741361/11088090\) | \(45416816640\) | \([4]\) | \(12288\) | \(1.4572\) | |
| 3120.w4 | 3120z6 | \([0, 1, 0, -1840, 834260]\) | \(-168288035761/73415764890\) | \(-300710972989440\) | \([4]\) | \(12288\) | \(1.4572\) |
Rank
sage: E.rank()
The elliptic curves in class 3120z have rank \(1\).
Complex multiplication
The elliptic curves in class 3120z do not have complex multiplication.Modular form 3120.2.a.z
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.
