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SageMath
E = EllipticCurve("ep1")
E.isogeny_class()
Elliptic curves in class 221760ep
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 221760.ci6 | 221760ep1 | \([0, 0, 0, -165043308, 816103020112]\) | \(2601656892010848045529/56330588160\) | \(10764943037206364160\) | \([2]\) | \(21233664\) | \(3.1788\) | \(\Gamma_0(N)\)-optimal |
| 221760.ci5 | 221760ep2 | \([0, 0, 0, -165227628, 814188820048]\) | \(2610383204210122997209/12104550027662400\) | \(2313215533427166963302400\) | \([2, 2]\) | \(42467328\) | \(3.5254\) | |
| 221760.ci4 | 221760ep3 | \([0, 0, 0, -176111148, 700405336528]\) | \(3160944030998056790089/720291785342976000\) | \(137649903767395894296576000\) | \([2]\) | \(63700992\) | \(3.7281\) | |
| 221760.ci7 | 221760ep4 | \([0, 0, 0, -81246828, 1641198146128]\) | \(-310366976336070130009/5909282337130963560\) | \(-1129281440649962438063554560\) | \([2]\) | \(84934656\) | \(3.8720\) | |
| 221760.ci3 | 221760ep5 | \([0, 0, 0, -252157548, -135329310128]\) | \(9278380528613437145689/5328033205714065000\) | \(1018203001838778026557440000\) | \([2]\) | \(84934656\) | \(3.8720\) | |
| 221760.ci2 | 221760ep6 | \([0, 0, 0, -931085868, -10336419100208]\) | \(467116778179943012100169/28800309694464000000\) | \(5503824892333721124864000000\) | \([2, 2]\) | \(127401984\) | \(4.0747\) | |
| 221760.ci8 | 221760ep7 | \([0, 0, 0, 727794132, -43161672988208]\) | \(223090928422700449019831/4340371122724101696000\) | \(-829457838497037061032247296000\) | \([2]\) | \(254803968\) | \(4.4213\) | |
| 221760.ci1 | 221760ep8 | \([0, 0, 0, -14669561388, -683867929163312]\) | \(1826870018430810435423307849/7641104625000000000\) | \(1460237833764864000000000000\) | \([2]\) | \(254803968\) | \(4.4213\) |
Rank
sage: E.rank()
The elliptic curves in class 221760ep have rank \(0\).
Complex multiplication
The elliptic curves in class 221760ep do not have complex multiplication.Modular form 221760.2.a.ep
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.
