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SageMath
E = EllipticCurve("br1")
E.isogeny_class()
Elliptic curves in class 39270.br
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 39270.br1 | 39270bp7 | \([1, 0, 1, -141721078503, -20534789449019414]\) | \(314794443646748303921433115102799635561/8206405838866889178408192798720\) | \(8206405838866889178408192798720\) | \([2]\) | \(207028224\) | \(5.0381\) | |
| 39270.br2 | 39270bp8 | \([1, 0, 1, -39084862183, 2681211362365418]\) | \(6603124212008881280120689341135103081/715642524575996594697670556160000\) | \(715642524575996594697670556160000\) | \([2]\) | \(207028224\) | \(5.0381\) | |
| 39270.br3 | 39270bp5 | \([1, 0, 1, -38005452808, 2851783142342918]\) | \(6071016954682394123338855607356153081/10029115297984535156250000\) | \(10029115297984535156250000\) | \([6]\) | \(69009408\) | \(4.4888\) | |
| 39270.br4 | 39270bp6 | \([1, 0, 1, -9201244903, -294611199823894]\) | \(86151626782508161683074667552941161/12360692761105045152384575078400\) | \(12360692761105045152384575078400\) | \([2, 2]\) | \(103514112\) | \(4.6915\) | |
| 39270.br5 | 39270bp4 | \([1, 0, 1, -3028263528, 18137793952006]\) | \(3071176032738522446354893004903161/1635177816170458876705577958000\) | \(1635177816170458876705577958000\) | \([6]\) | \(69009408\) | \(4.4888\) | |
| 39270.br6 | 39270bp2 | \([1, 0, 1, -2376073528, 44530097120006]\) | \(1483553933406627878314880715143161/1904972409734563785924000000\) | \(1904972409734563785924000000\) | \([2, 6]\) | \(34504704\) | \(4.1422\) | |
| 39270.br7 | 39270bp1 | \([1, 0, 1, -108475448, 1079289750278]\) | \(-141162084764748587904214427641/421539677967044903067648000\) | \(-421539677967044903067648000\) | \([6]\) | \(17252352\) | \(3.7957\) | \(\Gamma_0(N)\)-optimal |
| 39270.br8 | 39270bp3 | \([1, 0, 1, 948970777, -24855007825942]\) | \(94510971880619057444979349412759/321572798571266028122690027520\) | \(-321572798571266028122690027520\) | \([2]\) | \(51757056\) | \(4.3450\) |
Rank
sage: E.rank()
The elliptic curves in class 39270.br have rank \(1\).
Complex multiplication
The elliptic curves in class 39270.br do not have complex multiplication.Modular form 39270.2.a.br
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 4 & 12 & 2 & 3 & 6 & 12 & 4 \\ 4 & 1 & 3 & 2 & 12 & 6 & 12 & 4 \\ 12 & 3 & 1 & 6 & 4 & 2 & 4 & 12 \\ 2 & 2 & 6 & 1 & 6 & 3 & 6 & 2 \\ 3 & 12 & 4 & 6 & 1 & 2 & 4 & 12 \\ 6 & 6 & 2 & 3 & 2 & 1 & 2 & 6 \\ 12 & 12 & 4 & 6 & 4 & 2 & 1 & 3 \\ 4 & 4 & 12 & 2 & 12 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
