# Properties

 Label 10304.s Number of curves $4$ Conductor $10304$ CM no Rank $2$ Graph

# Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 10304.s

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
10304.s1 10304h3 $$[0, 0, 0, -7916, 271024]$$ $$209267191953/55223$$ $$14476378112$$ $$[2]$$ $$10240$$ $$0.93409$$
10304.s2 10304h2 $$[0, 0, 0, -556, 3120]$$ $$72511713/25921$$ $$6795034624$$ $$[2, 2]$$ $$5120$$ $$0.58751$$
10304.s3 10304h1 $$[0, 0, 0, -236, -1360]$$ $$5545233/161$$ $$42205184$$ $$[2]$$ $$2560$$ $$0.24094$$ $$\Gamma_0(N)$$-optimal
10304.s4 10304h4 $$[0, 0, 0, 1684, 21936]$$ $$2014698447/1958887$$ $$-513510473728$$ $$[2]$$ $$10240$$ $$0.93409$$

## Rank

sage: E.rank()

The elliptic curves in class 10304.s have rank $$2$$.

## Complex multiplication

The elliptic curves in class 10304.s do not have complex multiplication.

## Modular form 10304.2.a.s

sage: E.q_eigenform(10)

$$q - 2q^{5} + q^{7} - 3q^{9} - 4q^{11} - 6q^{13} - 2q^{17} - 4q^{19} + O(q^{20})$$

## 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 LMFDB numbering.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.