Show commands:
SageMath
E = EllipticCurve("he1")
E.isogeny_class()
Elliptic curves in class 33600.he
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
33600.he1 | 33600gt6 | \([0, 1, 0, -26880033, 53631540063]\) | \(524388516989299201/3150\) | \(12902400000000\) | \([2]\) | \(1179648\) | \(2.5812\) | |
33600.he2 | 33600gt4 | \([0, 1, 0, -1680033, 837540063]\) | \(128031684631201/9922500\) | \(40642560000000000\) | \([2, 2]\) | \(589824\) | \(2.2346\) | |
33600.he3 | 33600gt5 | \([0, 1, 0, -1568033, 954132063]\) | \(-104094944089921/35880468750\) | \(-146966400000000000000\) | \([2]\) | \(1179648\) | \(2.5812\) | |
33600.he4 | 33600gt3 | \([0, 1, 0, -592033, -165915937]\) | \(5602762882081/345888060\) | \(1416757493760000000\) | \([2]\) | \(589824\) | \(2.2346\) | |
33600.he5 | 33600gt2 | \([0, 1, 0, -112033, 11204063]\) | \(37966934881/8643600\) | \(35404185600000000\) | \([2, 2]\) | \(294912\) | \(1.8881\) | |
33600.he6 | 33600gt1 | \([0, 1, 0, 15967, 1092063]\) | \(109902239/188160\) | \(-770703360000000\) | \([2]\) | \(147456\) | \(1.5415\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 33600.he have rank \(0\).
Complex multiplication
The elliptic curves in class 33600.he do not have complex multiplication.Modular form 33600.2.a.he
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 4 & 8 \\ 2 & 1 & 2 & 4 & 2 & 4 \\ 4 & 2 & 1 & 8 & 4 & 8 \\ 8 & 4 & 8 & 1 & 2 & 4 \\ 4 & 2 & 4 & 2 & 1 & 2 \\ 8 & 4 & 8 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.