# Properties

 Label 311696i Number of curves $2$ Conductor $311696$ CM no Rank $1$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("i1")

sage: E.isogeny_class()

## Elliptic curves in class 311696i

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
311696.i2 311696i1 $$[0, 1, 0, 66752, 31247412]$$ $$4533086375/60669952$$ $$-440240213340454912$$ $$$$ $$3440640$$ $$2.0663$$ $$\Gamma_0(N)$$-optimal
311696.i1 311696i2 $$[0, 1, 0, -1172288, 456981556]$$ $$24553362849625/1755162752$$ $$12736011796872691712$$ $$$$ $$6881280$$ $$2.4128$$

## Rank

sage: E.rank()

The elliptic curves in class 311696i have rank $$1$$.

## Complex multiplication

The elliptic curves in class 311696i do not have complex multiplication.

## Modular form 311696.2.a.i

sage: E.q_eigenform(10)

$$q - 2q^{3} + q^{7} + q^{9} - 6q^{17} - 6q^{19} + O(q^{20})$$ ## 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.

$$\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels. 