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SageMath
E = EllipticCurve("bx1")
E.isogeny_class()
Elliptic curves in class 8190.bx
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
8190.bx1 | 8190bv8 | \([1, -1, 1, -20391107, 35440483331]\) | \(1286229821345376481036009/247265484375000000\) | \(180256538109375000000\) | \([12]\) | \(663552\) | \(2.8872\) | |
8190.bx2 | 8190bv7 | \([1, -1, 1, -8969027, -10011937021]\) | \(109454124781830273937129/3914078300576808000\) | \(2853363081120493032000\) | \([6]\) | \(663552\) | \(2.8872\) | |
8190.bx3 | 8190bv4 | \([1, -1, 1, -8890592, -10201158529]\) | \(106607603143751752938169/5290068420\) | \(3856459878180\) | \([2]\) | \(221184\) | \(2.3379\) | |
8190.bx4 | 8190bv6 | \([1, -1, 1, -1409027, 429934979]\) | \(424378956393532177129/136231857216000000\) | \(99313023910464000000\) | \([2, 6]\) | \(331776\) | \(2.5407\) | |
8190.bx5 | 8190bv5 | \([1, -1, 1, -618872, -120756481]\) | \(35958207000163259449/12145729518877500\) | \(8854236819261697500\) | \([4]\) | \(221184\) | \(2.3379\) | |
8190.bx6 | 8190bv2 | \([1, -1, 1, -555692, -159271009]\) | \(26031421522845051769/5797789779600\) | \(4226588749328400\) | \([2, 2]\) | \(110592\) | \(1.9914\) | |
8190.bx7 | 8190bv1 | \([1, -1, 1, -30812, -3066721]\) | \(-4437543642183289/3033210136320\) | \(-2211210189377280\) | \([2]\) | \(55296\) | \(1.6448\) | \(\Gamma_0(N)\)-optimal |
8190.bx8 | 8190bv3 | \([1, -1, 1, 249853, 45738371]\) | \(2366200373628880151/2612420149248000\) | \(-1904454288801792000\) | \([6]\) | \(165888\) | \(2.1941\) |
Rank
sage: E.rank()
The elliptic curves in class 8190.bx have rank \(0\).
Complex multiplication
The elliptic curves in class 8190.bx do not have complex multiplication.Modular form 8190.2.a.bx
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 4 & 12 & 2 & 3 & 6 & 12 & 4 \\ 4 & 1 & 3 & 2 & 12 & 6 & 12 & 4 \\ 12 & 3 & 1 & 6 & 4 & 2 & 4 & 12 \\ 2 & 2 & 6 & 1 & 6 & 3 & 6 & 2 \\ 3 & 12 & 4 & 6 & 1 & 2 & 4 & 12 \\ 6 & 6 & 2 & 3 & 2 & 1 & 2 & 6 \\ 12 & 12 & 4 & 6 & 4 & 2 & 1 & 3 \\ 4 & 4 & 12 & 2 & 12 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.