# Properties

 Label 111600ev Number of curves $6$ Conductor $111600$ CM no Rank $2$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("111600.de1")

sage: E.isogeny_class()

## Elliptic curves in class 111600ev

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
111600.de6 111600ev1 [0, 0, 0, 215925, -218807750] [2] 2359296 $$\Gamma_0(N)$$-optimal
111600.de5 111600ev2 [0, 0, 0, -4392075, -3347639750] [2, 2] 4718592
111600.de4 111600ev3 [0, 0, 0, -13320075, 14588712250] [2] 9437184
111600.de2 111600ev4 [0, 0, 0, -69192075, -221529239750] [2, 2] 9437184
111600.de3 111600ev5 [0, 0, 0, -68112075, -228779279750] [4] 18874368
111600.de1 111600ev6 [0, 0, 0, -1107072075, -14177901599750] [2] 18874368

## Rank

sage: E.rank()

The elliptic curves in class 111600ev have rank $$2$$.

## Modular form 111600.2.a.de

sage: E.q_eigenform(10)

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

## 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.