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SageMath
E = EllipticCurve("dz1")
E.isogeny_class()
Elliptic curves in class 227430.dz
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 227430.dz1 | 227430cf8 | \([1, -1, 1, -6240679268, -189754610782719]\) | \(783736670177727068275201/360150\) | \(12351865476727350\) | \([2]\) | \(113246208\) | \(3.8160\) | |
| 227430.dz2 | 227430cf6 | \([1, -1, 1, -390042518, -2964841647519]\) | \(191342053882402567201/129708022500\) | \(4448524351443355102500\) | \([2, 2]\) | \(56623104\) | \(3.4694\) | |
| 227430.dz3 | 227430cf7 | \([1, -1, 1, -387605768, -3003716582319]\) | \(-187778242790732059201/4984939585440150\) | \(-170965717531500032921647350\) | \([2]\) | \(113246208\) | \(3.8160\) | |
| 227430.dz4 | 227430cf3 | \([1, -1, 1, -48962498, 131845180497]\) | \(378499465220294881/120530818800\) | \(4133778868852977481200\) | \([2]\) | \(28311552\) | \(3.1228\) | |
| 227430.dz5 | 227430cf4 | \([1, -1, 1, -24530018, -45712617519]\) | \(47595748626367201/1215506250000\) | \(41687545983954806250000\) | \([2, 2]\) | \(28311552\) | \(3.1228\) | |
| 227430.dz6 | 227430cf2 | \([1, -1, 1, -3476498, 1464110097]\) | \(135487869158881/51438240000\) | \(1764148884741401760000\) | \([2, 2]\) | \(14155776\) | \(2.7762\) | |
| 227430.dz7 | 227430cf1 | \([1, -1, 1, 682222, 163262481]\) | \(1023887723039/928972800\) | \(-31860466630955827200\) | \([2]\) | \(7077888\) | \(2.4297\) | \(\Gamma_0(N)\)-optimal |
| 227430.dz8 | 227430cf5 | \([1, -1, 1, 4126162, -146146797183]\) | \(226523624554079/269165039062500\) | \(-9231404563482055664062500\) | \([2]\) | \(56623104\) | \(3.4694\) |
Rank
sage: E.rank()
The elliptic curves in class 227430.dz have rank \(0\).
Complex multiplication
The elliptic curves in class 227430.dz do not have complex multiplication.Modular form 227430.2.a.dz
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 16 & 4 & 8 & 16 & 8 \\ 2 & 1 & 2 & 8 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 16 & 4 & 8 & 16 & 8 \\ 16 & 8 & 16 & 1 & 4 & 2 & 4 & 8 \\ 4 & 2 & 4 & 4 & 1 & 2 & 4 & 2 \\ 8 & 4 & 8 & 2 & 2 & 1 & 2 & 4 \\ 16 & 8 & 16 & 4 & 4 & 2 & 1 & 8 \\ 8 & 4 & 8 & 8 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
