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SageMath
E = EllipticCurve("de1")
E.isogeny_class()
Elliptic curves in class 111600.de
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 111600.de1 | 111600ev6 | \([0, 0, 0, -1107072075, -14177901599750]\) | \(3216206300355197383681/57660\) | \(2690184960000000\) | \([2]\) | \(18874368\) | \(3.4290\) | |
| 111600.de2 | 111600ev4 | \([0, 0, 0, -69192075, -221529239750]\) | \(785209010066844481/3324675600\) | \(155116064793600000000\) | \([2, 2]\) | \(9437184\) | \(3.0824\) | |
| 111600.de3 | 111600ev5 | \([0, 0, 0, -68112075, -228779279750]\) | \(-749011598724977281/51173462246460\) | \(-2387549054570837760000000\) | \([4]\) | \(18874368\) | \(3.4290\) | |
| 111600.de4 | 111600ev3 | \([0, 0, 0, -13320075, 14588712250]\) | \(5601911201812801/1271193750000\) | \(59308815600000000000000\) | \([2]\) | \(9437184\) | \(3.0824\) | |
| 111600.de5 | 111600ev2 | \([0, 0, 0, -4392075, -3347639750]\) | \(200828550012481/12454560000\) | \(581079951360000000000\) | \([2, 2]\) | \(4718592\) | \(2.7359\) | |
| 111600.de6 | 111600ev1 | \([0, 0, 0, 215925, -218807750]\) | \(23862997439/457113600\) | \(-21327092121600000000\) | \([2]\) | \(2359296\) | \(2.3893\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 111600.de have rank \(2\).
Complex multiplication
The elliptic curves in class 111600.de do not have complex multiplication.Modular form 111600.2.a.de
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 4 & 8 \\ 2 & 1 & 2 & 4 & 2 & 4 \\ 4 & 2 & 1 & 8 & 4 & 8 \\ 8 & 4 & 8 & 1 & 2 & 4 \\ 4 & 2 & 4 & 2 & 1 & 2 \\ 8 & 4 & 8 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
