# Properties

 Label 12696j Number of curves $6$ Conductor $12696$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("j1")

sage: E.isogeny_class()

## Elliptic curves in class 12696j

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
12696.k5 12696j1 $$[0, -1, 0, 353, -3272]$$ $$2048/3$$ $$-7105722672$$ $$[2]$$ $$5632$$ $$0.57582$$ $$\Gamma_0(N)$$-optimal
12696.k4 12696j2 $$[0, -1, 0, -2292, -30780]$$ $$35152/9$$ $$341074688256$$ $$[2, 2]$$ $$11264$$ $$0.92239$$
12696.k2 12696j3 $$[0, -1, 0, -34032, -2404932]$$ $$28756228/3$$ $$454766251008$$ $$[2]$$ $$22528$$ $$1.2690$$
12696.k3 12696j4 $$[0, -1, 0, -12872, 540540]$$ $$1556068/81$$ $$12278688777216$$ $$[2, 2]$$ $$22528$$ $$1.2690$$
12696.k1 12696j5 $$[0, -1, 0, -203312, 35352972]$$ $$3065617154/9$$ $$2728597506048$$ $$[2]$$ $$45056$$ $$1.6155$$
12696.k6 12696j6 $$[0, -1, 0, 8288, 2123308]$$ $$207646/6561$$ $$-1989147581908992$$ $$[2]$$ $$45056$$ $$1.6155$$

## Rank

sage: E.rank()

The elliptic curves in class 12696j have rank $$1$$.

## Complex multiplication

The elliptic curves in class 12696j do not have complex multiplication.

## Modular form 12696.2.a.j

sage: E.q_eigenform(10)

$$q - q^{3} + 2q^{5} + q^{9} - 4q^{11} - 2q^{13} - 2q^{15} - 2q^{17} + 4q^{19} + O(q^{20})$$

## 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.