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SageMath
E = EllipticCurve("dg1")
E.isogeny_class()
Elliptic curves in class 123210.dg
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 123210.dg1 | 123210dh4 | \([1, -1, 1, -4201522862, -104822341364739]\) | \(4385367890843575421521/24975000000\) | \(46713603440220975000000\) | \([2]\) | \(75644928\) | \(3.9616\) | |
| 123210.dg2 | 123210dh6 | \([1, -1, 1, -3734803382, 87470606300829]\) | \(3080272010107543650001/15465841417699560\) | \(28927534849079567856246749160\) | \([2]\) | \(151289856\) | \(4.3082\) | |
| 123210.dg3 | 123210dh3 | \([1, -1, 1, -361313582, -296803127811]\) | \(2788936974993502801/1593609593601600\) | \(2980710574335809875053057600\) | \([2, 2]\) | \(75644928\) | \(3.9616\) | |
| 123210.dg4 | 123210dh2 | \([1, -1, 1, -262745582, -1635829694211]\) | \(1072487167529950801/2554882560000\) | \(4778689519286349212160000\) | \([2, 2]\) | \(37822464\) | \(3.6150\) | |
| 123210.dg5 | 123210dh1 | \([1, -1, 1, -10411502, -44510052099]\) | \(-66730743078481/419010969600\) | \(-783724215054930385305600\) | \([4]\) | \(18911232\) | \(3.2684\) | \(\Gamma_0(N)\)-optimal |
| 123210.dg6 | 123210dh5 | \([1, -1, 1, 1435088218, -2367695122851]\) | \(174751791402194852399/102423900876336360\) | \(-191575154688195328231800873960\) | \([2]\) | \(151289856\) | \(4.3082\) |
Rank
sage: E.rank()
The elliptic curves in class 123210.dg have rank \(0\).
Complex multiplication
The elliptic curves in class 123210.dg do not have complex multiplication.Modular form 123210.2.a.dg
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 4 & 2 & 4 & 8 \\ 8 & 1 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 2 & 4 & 2 & 1 & 2 & 4 \\ 4 & 8 & 4 & 2 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
