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SageMath
E = EllipticCurve("iz1")
E.isogeny_class()
Elliptic curves in class 308550iz
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
308550.iz6 | 308550iz1 | \([1, 0, 0, -242063, -56341383]\) | \(-56667352321/16711680\) | \(-462590008320000000\) | \([2]\) | \(3932160\) | \(2.1041\) | \(\Gamma_0(N)\)-optimal |
308550.iz5 | 308550iz2 | \([1, 0, 0, -4114063, -3212021383]\) | \(278202094583041/16646400\) | \(460783016100000000\) | \([2, 2]\) | \(7864320\) | \(2.4507\) | |
308550.iz4 | 308550iz3 | \([1, 0, 0, -4356063, -2812963383]\) | \(330240275458561/67652010000\) | \(1872650976368906250000\) | \([2, 2]\) | \(15728640\) | \(2.7973\) | |
308550.iz2 | 308550iz4 | \([1, 0, 0, -65824063, -205559111383]\) | \(1139466686381936641/4080\) | \(112937013750000\) | \([2]\) | \(15728640\) | \(2.7973\) | |
308550.iz3 | 308550iz5 | \([1, 0, 0, -21840563, 36789429117]\) | \(41623544884956481/2962701562500\) | \(82009477230688476562500\) | \([2, 2]\) | \(31457280\) | \(3.1438\) | |
308550.iz7 | 308550iz6 | \([1, 0, 0, 9256437, -16874675883]\) | \(3168685387909439/6278181696900\) | \(-173784091330341576562500\) | \([2]\) | \(31457280\) | \(3.1438\) | |
308550.iz1 | 308550iz7 | \([1, 0, 0, -343246813, 2447657710367]\) | \(161572377633716256481/914742821250\) | \(25320667299319863281250\) | \([2]\) | \(62914560\) | \(3.4904\) | |
308550.iz8 | 308550iz8 | \([1, 0, 0, 19813687, 160544205867]\) | \(31077313442863199/420227050781250\) | \(-11632153973579406738281250\) | \([2]\) | \(62914560\) | \(3.4904\) |
Rank
sage: E.rank()
The elliptic curves in class 308550iz have rank \(1\).
Complex multiplication
The elliptic curves in class 308550iz do not have complex multiplication.Modular form 308550.2.a.iz
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 8 & 16 & 16 \\ 2 & 1 & 2 & 2 & 4 & 4 & 8 & 8 \\ 4 & 2 & 1 & 4 & 2 & 2 & 4 & 4 \\ 4 & 2 & 4 & 1 & 8 & 8 & 16 & 16 \\ 8 & 4 & 2 & 8 & 1 & 4 & 2 & 2 \\ 8 & 4 & 2 & 8 & 4 & 1 & 8 & 8 \\ 16 & 8 & 4 & 16 & 2 & 8 & 1 & 4 \\ 16 & 8 & 4 & 16 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.