# Properties

 Label 235200gn Number of curves 8 Conductor 235200 CM no Rank 0 Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("235200.gn1")

sage: E.isogeny_class()

## Elliptic curves in class 235200gn

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
235200.gn7 235200gn1 [0, -1, 0, -39005633, -93715660863] [2] 21233664 $$\Gamma_0(N)$$-optimal
235200.gn6 235200gn2 [0, -1, 0, -45277633, -61546572863] [2, 2] 42467328
235200.gn5 235200gn3 [0, -1, 0, -115445633, 362883979137] [2] 63700992
235200.gn4 235200gn4 [0, -1, 0, -341629633, 2387802707137] [2] 84934656
235200.gn8 235200gn5 [0, -1, 0, 150722367, -452174572863] [2] 84934656
235200.gn2 235200gn6 [0, -1, 0, -1721077633, 27480402827137] [2, 2] 127401984
235200.gn1 235200gn7 [0, -1, 0, -27536629633, 1758800397707137] [2] 254803968
235200.gn3 235200gn8 [0, -1, 0, -1595637633, 31655924107137] [2] 254803968

## Rank

sage: E.rank()

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

## Modular form 235200.2.a.gn

sage: E.q_eigenform(10)

$$q - q^{3} + q^{9} - 2q^{13} - 6q^{17} + 8q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels.