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SageMath
E = EllipticCurve("df1")
E.isogeny_class()
Elliptic curves in class 182070.df
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
182070.df1 | 182070v7 | \([1, -1, 1, -913555787, 10628208252699]\) | \(4791901410190533590281/41160000\) | \(724263205889160000\) | \([2]\) | \(42467328\) | \(3.4682\) | |
182070.df2 | 182070v6 | \([1, -1, 1, -57098507, 166068703131]\) | \(1169975873419524361/108425318400\) | \(1907883107481460838400\) | \([2, 2]\) | \(21233664\) | \(3.1216\) | |
182070.df3 | 182070v8 | \([1, -1, 1, -52936907, 191299651611]\) | \(-932348627918877961/358766164249920\) | \(-6312952679402429611225920\) | \([2]\) | \(42467328\) | \(3.4682\) | |
182070.df4 | 182070v4 | \([1, -1, 1, -11333912, 14431100199]\) | \(9150443179640281/184570312500\) | \(3247752338270507812500\) | \([2]\) | \(14155776\) | \(2.9189\) | |
182070.df5 | 182070v3 | \([1, -1, 1, -3830027, 2193551259]\) | \(353108405631241/86318776320\) | \(1518890030756863672320\) | \([2]\) | \(10616832\) | \(2.7750\) | |
182070.df6 | 182070v2 | \([1, -1, 1, -1502132, -371627769]\) | \(21302308926361/8930250000\) | \(157139249134880250000\) | \([2, 2]\) | \(7077888\) | \(2.5723\) | |
182070.df7 | 182070v1 | \([1, -1, 1, -1294052, -566057721]\) | \(13619385906841/6048000\) | \(106422348620448000\) | \([2]\) | \(3538944\) | \(2.2257\) | \(\Gamma_0(N)\)-optimal |
182070.df8 | 182070v5 | \([1, -1, 1, 5000368, -2733335769]\) | \(785793873833639/637994920500\) | \(-11226342236694114820500\) | \([2]\) | \(14155776\) | \(2.9189\) |
Rank
sage: E.rank()
The elliptic curves in class 182070.df have rank \(0\).
Complex multiplication
The elliptic curves in class 182070.df do not have complex multiplication.Modular form 182070.2.a.df
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 3 & 4 & 6 & 12 & 12 \\ 2 & 1 & 2 & 6 & 2 & 3 & 6 & 6 \\ 4 & 2 & 1 & 12 & 4 & 6 & 12 & 3 \\ 3 & 6 & 12 & 1 & 12 & 2 & 4 & 4 \\ 4 & 2 & 4 & 12 & 1 & 6 & 3 & 12 \\ 6 & 3 & 6 & 2 & 6 & 1 & 2 & 2 \\ 12 & 6 & 12 & 4 & 3 & 2 & 1 & 4 \\ 12 & 6 & 3 & 4 & 12 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.