Properties

 Label 235200.gk Number of curves $8$ Conductor $235200$ CM no Rank $0$ Graph

Related objects

Show commands for: SageMath
sage: E = EllipticCurve("235200.gk1")

sage: E.isogeny_class()

Elliptic curves in class 235200.gk

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
235200.gk1 235200gk8 [0, -1, 0, -505760033, -2754117780063] [2] 127401984
235200.gk2 235200gk5 [0, -1, 0, -451664033, -3694482660063] [2] 42467328
235200.gk3 235200gk6 [0, -1, 0, -211760033, 1154612219937] [2, 2] 63700992
235200.gk4 235200gk3 [0, -1, 0, -210192033, 1173000155937] [2] 31850496
235200.gk5 235200gk2 [0, -1, 0, -28304033, -57396900063] [2, 2] 21233664
235200.gk6 235200gk4 [0, -1, 0, -6352033, -144173156063] [2] 42467328
235200.gk7 235200gk1 [0, -1, 0, -3216033, 782171937] [2] 10616832 $$\Gamma_0(N)$$-optimal
235200.gk8 235200gk7 [0, -1, 0, 57151967, 3886489227937] [2] 127401984

Rank

sage: E.rank()

The elliptic curves in class 235200.gk have rank $$0$$.

Modular form 235200.2.a.gk

sage: E.q_eigenform(10)

$$q - q^{3} + q^{9} - 2q^{13} - 6q^{17} - 4q^{19} + O(q^{20})$$

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.

$$\left(\begin{array}{rrrrrrrr} 1 & 3 & 2 & 4 & 6 & 12 & 12 & 4 \\ 3 & 1 & 6 & 12 & 2 & 4 & 4 & 12 \\ 2 & 6 & 1 & 2 & 3 & 6 & 6 & 2 \\ 4 & 12 & 2 & 1 & 6 & 12 & 3 & 4 \\ 6 & 2 & 3 & 6 & 1 & 2 & 2 & 6 \\ 12 & 4 & 6 & 12 & 2 & 1 & 4 & 3 \\ 12 & 4 & 6 & 3 & 2 & 4 & 1 & 12 \\ 4 & 12 & 2 & 4 & 6 & 3 & 12 & 1 \end{array}\right)$$

Isogeny graph

sage: E.isogeny_graph().plot(edge_labels=True)

The vertices are labelled with LMFDB labels.