# Properties

 Label 5712.q Number of curves $4$ Conductor $5712$ CM no Rank $1$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("q1")

sage: E.isogeny_class()

## Elliptic curves in class 5712.q

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
5712.q1 5712bb3 $$[0, 1, 0, -81264, -8943660]$$ $$14489843500598257/6246072$$ $$25583910912$$ $$$$ $$18432$$ $$1.3397$$
5712.q2 5712bb4 $$[0, 1, 0, -10864, 226772]$$ $$34623662831857/14438442312$$ $$59139859709952$$ $$$$ $$18432$$ $$1.3397$$
5712.q3 5712bb2 $$[0, 1, 0, -5104, -139564]$$ $$3590714269297/73410624$$ $$300689915904$$ $$[2, 2]$$ $$9216$$ $$0.99312$$
5712.q4 5712bb1 $$[0, 1, 0, 16, -6444]$$ $$103823/4386816$$ $$-17968398336$$ $$$$ $$4608$$ $$0.64655$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 5712.q have rank $$1$$.

## Complex multiplication

The elliptic curves in class 5712.q do not have complex multiplication.

## Modular form5712.2.a.q

sage: E.q_eigenform(10)

$$q + q^{3} - 2q^{5} + q^{7} + q^{9} - 6q^{13} - 2q^{15} + q^{17} + O(q^{20})$$ ## 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 LMFDB numbering.

$$\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 