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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 106470.a
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 106470.a1 | 106470bn5 | \([1, -1, 0, -1296308025, 17964217463661]\) | \(68463752473882049153689/1817088000000000\) | \(6393867063187968000000000\) | \([2]\) | \(62705664\) | \(3.8650\) | |
| 106470.a2 | 106470bn6 | \([1, -1, 0, -1245689145, 19431547433325]\) | \(-60752633741424905775769/11197265625000000000\) | \(-39400308558228515625000000000\) | \([2]\) | \(125411328\) | \(4.2116\) | |
| 106470.a3 | 106470bn3 | \([1, -1, 0, -27771210, -16125914700]\) | \(673163386034885929/357608625192000\) | \(1258333118774137427112000\) | \([2]\) | \(20901888\) | \(3.3157\) | |
| 106470.a4 | 106470bn1 | \([1, -1, 0, -21816495, -39216122919]\) | \(326355561310674169/465699780\) | \(1638678195374072580\) | \([2]\) | \(6967296\) | \(2.7664\) | \(\Gamma_0(N)\)-optimal |
| 106470.a5 | 106470bn2 | \([1, -1, 0, -21618765, -39962000025]\) | \(-317562142497484249/12339342574650\) | \(-43418984699391364258650\) | \([2]\) | \(13934592\) | \(3.1130\) | |
| 106470.a6 | 106470bn4 | \([1, -1, 0, 105894270, -126239537124]\) | \(37321015309599759191/23553520979625000\) | \(-82878804996638076869625000\) | \([2]\) | \(41803776\) | \(3.6623\) |
Rank
sage: E.rank()
The elliptic curves in class 106470.a have rank \(0\).
Complex multiplication
The elliptic curves in class 106470.a do not have complex multiplication.Modular form 106470.2.a.a
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 3 & 9 & 18 & 6 \\ 2 & 1 & 6 & 18 & 9 & 3 \\ 3 & 6 & 1 & 3 & 6 & 2 \\ 9 & 18 & 3 & 1 & 2 & 6 \\ 18 & 9 & 6 & 2 & 1 & 3 \\ 6 & 3 & 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
