Show commands:
SageMath
E = EllipticCurve("hj1")
E.isogeny_class()
Elliptic curves in class 129360.hj
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 129360.hj1 | 129360hy8 | \([0, 1, 0, -19966903000, -1085963598703852]\) | \(1826870018430810435423307849/7641104625000000000\) | \(3682174230637056000000000000\) | \([2]\) | \(191102976\) | \(4.4983\) | |
| 129360.hj2 | 129360hy6 | \([0, 1, 0, -1267311320, -16414273137900]\) | \(467116778179943012100169/28800309694464000000\) | \(13878589993959404077056000000\) | \([2, 2]\) | \(95551488\) | \(4.1518\) | |
| 129360.hj3 | 129360hy5 | \([0, 1, 0, -343214440, -215012337100]\) | \(9278380528613437145689/5328033205714065000\) | \(2567527541223645319925760000\) | \([2]\) | \(63700992\) | \(3.9490\) | |
| 129360.hj4 | 129360hy3 | \([0, 1, 0, -239706840, 1112137831188]\) | \(3160944030998056790089/720291785342976000\) | \(347101627407629448904704000\) | \([2]\) | \(47775744\) | \(3.8052\) | |
| 129360.hj5 | 129360hy2 | \([0, 1, 0, -224893160, 1292826726708]\) | \(2610383204210122997209/12104550027662400\) | \(5833065292613442345369600\) | \([2, 2]\) | \(31850496\) | \(3.6025\) | |
| 129360.hj6 | 129360hy1 | \([0, 1, 0, -224642280, 1295866489140]\) | \(2601656892010848045529/56330588160\) | \(27145164252921200640\) | \([2]\) | \(15925248\) | \(3.2559\) | \(\Gamma_0(N)\)-optimal |
| 129360.hj7 | 129360hy4 | \([0, 1, 0, -110585960, 2606125008948]\) | \(-310366976336070130009/5909282337130963560\) | \(-2847625861861870517741322240\) | \([2]\) | \(63700992\) | \(3.9490\) | |
| 129360.hj8 | 129360hy7 | \([0, 1, 0, 990608680, -68538807921900]\) | \(223090928422700449019831/4340371122724101696000\) | \(-2091582759802338674412355584000\) | \([2]\) | \(191102976\) | \(4.4983\) |
Rank
sage: E.rank()
The elliptic curves in class 129360.hj have rank \(0\).
Complex multiplication
The elliptic curves in class 129360.hj do not have complex multiplication.Modular form 129360.2.a.hj
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 6 & 12 & 12 & 4 \\ 2 & 1 & 6 & 2 & 3 & 6 & 6 & 2 \\ 3 & 6 & 1 & 12 & 2 & 4 & 4 & 12 \\ 4 & 2 & 12 & 1 & 6 & 3 & 12 & 4 \\ 6 & 3 & 2 & 6 & 1 & 2 & 2 & 6 \\ 12 & 6 & 4 & 3 & 2 & 1 & 4 & 12 \\ 12 & 6 & 4 & 12 & 2 & 4 & 1 & 3 \\ 4 & 2 & 12 & 4 & 6 & 12 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
