# Properties

 Label 2800s Number of curves $3$ Conductor $2800$ CM no Rank $1$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("s1")

sage: E.isogeny_class()

## Elliptic curves in class 2800s

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
2800.z2 2800s1 $$[0, 1, 0, -533, -5437]$$ $$-262144/35$$ $$-2240000000$$ $$[]$$ $$1152$$ $$0.52672$$ $$\Gamma_0(N)$$-optimal
2800.z3 2800s2 $$[0, 1, 0, 3467, 14563]$$ $$71991296/42875$$ $$-2744000000000$$ $$[]$$ $$3456$$ $$1.0760$$
2800.z1 2800s3 $$[0, 1, 0, -52533, 4830563]$$ $$-250523582464/13671875$$ $$-875000000000000$$ $$[]$$ $$10368$$ $$1.6253$$

## Rank

sage: E.rank()

The elliptic curves in class 2800s have rank $$1$$.

## Complex multiplication

The elliptic curves in class 2800s do not have complex multiplication.

## Modular form2800.2.a.s

sage: E.q_eigenform(10)

$$q + q^{3} + q^{7} - 2q^{9} + 3q^{11} - 5q^{13} - 3q^{17} - 2q^{19} + O(q^{20})$$ ## 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.

$$\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels. 