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SageMath
E = EllipticCurve("el1")
E.isogeny_class()
Elliptic curves in class 324870el
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 324870.el7 | 324870el1 | \([1, 1, 1, -102711155, -385844310223]\) | \(1018563973439611524445729/42904970360310988800\) | \(5047726857920227521331200\) | \([2]\) | \(79626240\) | \(3.5047\) | \(\Gamma_0(N)\)-optimal |
| 324870.el6 | 324870el2 | \([1, 1, 1, -272306035, 1216420278065]\) | \(18980483520595353274840609/5549773448629762560000\) | \(652925296457842935421440000\) | \([2, 2]\) | \(159252480\) | \(3.8513\) | |
| 324870.el5 | 324870el3 | \([1, 1, 1, -1263846515, 17180605977905]\) | \(1897660325010178513043539489/14258428094958372000000\) | \(1677489806943757507428000000\) | \([2]\) | \(238878720\) | \(4.0540\) | |
| 324870.el4 | 324870el4 | \([1, 1, 1, -3987290035, 96895090201265]\) | \(59589391972023341137821784609/8834417507562311995200\) | \(1039360385347198443923284800\) | \([2]\) | \(318504960\) | \(4.1978\) | |
| 324870.el8 | 324870el5 | \([1, 1, 1, 729159885, 8086075902897]\) | \(364421318680576777174674911/450962301637624725000000\) | \(-53055263825364911271525000000\) | \([2]\) | \(318504960\) | \(4.1978\) | |
| 324870.el2 | 324870el6 | \([1, 1, 1, -20184937795, 1103788604442257]\) | \(7730680381889320597382223137569/441370202660156250000\) | \(51926762972764722656250000\) | \([2, 2]\) | \(477757440\) | \(4.4006\) | |
| 324870.el1 | 324870el7 | \([1, 1, 1, -322959000295, 70642876415817257]\) | \(31664865542564944883878115208137569/103216295812500\) | \(12143293986044812500\) | \([2]\) | \(955514880\) | \(4.7472\) | |
| 324870.el3 | 324870el8 | \([1, 1, 1, -20148335775, 1107991116611385]\) | \(-7688701694683937879808871873249/58423707246780395507812500\) | \(-6873490733876466751098632812500\) | \([2]\) | \(955514880\) | \(4.7472\) |
Rank
sage: E.rank()
The elliptic curves in class 324870el have rank \(0\).
Complex multiplication
The elliptic curves in class 324870el do not have complex multiplication.Modular form 324870.2.a.el
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 Cremona numbering.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
