Properties

 Label 25200dx Number of curves 6 Conductor 25200 CM no Rank 1 Graph

Related objects

Show commands for: SageMath
sage: E = EllipticCurve("25200.cr1")

sage: E.isogeny_class()

Elliptic curves in class 25200dx

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
25200.cr6 25200dx1 [0, 0, 0, 3525, -17750] [2] 32768 $$\Gamma_0(N)$$-optimal
25200.cr5 25200dx2 [0, 0, 0, -14475, -143750] [2, 2] 65536
25200.cr3 25200dx3 [0, 0, 0, -140475, 20142250] [2] 131072
25200.cr2 25200dx4 [0, 0, 0, -176475, -28493750] [2, 2] 131072
25200.cr4 25200dx5 [0, 0, 0, -122475, -46259750] [2] 262144
25200.cr1 25200dx6 [0, 0, 0, -2822475, -1825127750] [2] 262144

Rank

sage: E.rank()

The elliptic curves in class 25200dx have rank $$1$$.

Modular form 25200.2.a.cr

sage: E.q_eigenform(10)

$$q - q^{7} + 4q^{11} + 2q^{13} - 6q^{17} - 4q^{19} + O(q^{20})$$

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 Cremona numbering.

$$\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)$$

Isogeny graph

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

The vertices are labelled with Cremona labels.