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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 1170.a
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 1170.a1 | 1170d5 | \([1, -1, 0, -81135, 8915611]\) | \(81025909800741361/11088090\) | \(8083217610\) | \([2]\) | \(4096\) | \(1.3134\) | |
| 1170.a2 | 1170d3 | \([1, -1, 0, -7605, -253175]\) | \(66730743078481/60937500\) | \(44423437500\) | \([2]\) | \(2048\) | \(0.96681\) | |
| 1170.a3 | 1170d4 | \([1, -1, 0, -5085, 139441]\) | \(19948814692561/231344100\) | \(168649848900\) | \([2, 2]\) | \(2048\) | \(0.96681\) | |
| 1170.a4 | 1170d6 | \([1, -1, 0, -1035, 352471]\) | \(-168288035761/73415764890\) | \(-53520092604810\) | \([2]\) | \(4096\) | \(1.3134\) | |
| 1170.a5 | 1170d2 | \([1, -1, 0, -585, -1859]\) | \(30400540561/15210000\) | \(11088090000\) | \([2, 2]\) | \(1024\) | \(0.62023\) | |
| 1170.a6 | 1170d1 | \([1, -1, 0, 135, -275]\) | \(371694959/249600\) | \(-181958400\) | \([2]\) | \(512\) | \(0.27366\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 1170.a have rank \(1\).
Complex multiplication
The elliptic curves in class 1170.a do not have complex multiplication.Modular form 1170.2.a.a
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 LMFDB numbering.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 2 & 4 & 4 & 8 \\ 8 & 1 & 4 & 8 & 2 & 4 \\ 2 & 4 & 1 & 2 & 2 & 4 \\ 4 & 8 & 2 & 1 & 4 & 8 \\ 4 & 2 & 2 & 4 & 1 & 2 \\ 8 & 4 & 4 & 8 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
