# Properties

 Label 3136y Number of curves $6$ Conductor $3136$ CM no Rank $1$ Graph

# Related objects

Show commands: SageMath
E = EllipticCurve("y1")

E.isogeny_class()

## Elliptic curves in class 3136y

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
3136.z5 3136y1 $$[0, -1, 0, -1633, 51969]$$ $$-15625/28$$ $$-863547424768$$ $$[2]$$ $$3072$$ $$0.98059$$ $$\Gamma_0(N)$$-optimal
3136.z4 3136y2 $$[0, -1, 0, -32993, 2316161]$$ $$128787625/98$$ $$3022415986688$$ $$[2]$$ $$6144$$ $$1.3272$$
3136.z6 3136y3 $$[0, -1, 0, 14047, -1080127]$$ $$9938375/21952$$ $$-677021181018112$$ $$[2]$$ $$9216$$ $$1.5299$$
3136.z3 3136y4 $$[0, -1, 0, -111393, -11692351]$$ $$4956477625/941192$$ $$29027283136151552$$ $$[2]$$ $$18432$$ $$1.8765$$
3136.z2 3136y5 $$[0, -1, 0, -534753, -150770815]$$ $$-548347731625/1835008$$ $$-56593444029595648$$ $$[2]$$ $$27648$$ $$2.0792$$
3136.z1 3136y6 $$[0, -1, 0, -8562913, -9641661567]$$ $$2251439055699625/25088$$ $$773738492592128$$ $$[2]$$ $$55296$$ $$2.4258$$

## Rank

sage: E.rank()

The elliptic curves in class 3136y have rank $$1$$.

## Complex multiplication

The elliptic curves in class 3136y do not have complex multiplication.

## Modular form3136.2.a.y

sage: E.q_eigenform(10)

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

## 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.

$$\left(\begin{array}{rrrrrr} 1 & 2 & 3 & 6 & 9 & 18 \\ 2 & 1 & 6 & 3 & 18 & 9 \\ 3 & 6 & 1 & 2 & 3 & 6 \\ 6 & 3 & 2 & 1 & 6 & 3 \\ 9 & 18 & 3 & 6 & 1 & 2 \\ 18 & 9 & 6 & 3 & 2 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels.