# Properties

 Label 10304.r Number of curves $4$ Conductor $10304$ CM no Rank $1$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("r1")

sage: E.isogeny_class()

## Elliptic curves in class 10304.r

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
10304.r1 10304w3 $$[0, 0, 0, -7916, -271024]$$ $$209267191953/55223$$ $$14476378112$$ $$$$ $$10240$$ $$0.93409$$
10304.r2 10304w2 $$[0, 0, 0, -556, -3120]$$ $$72511713/25921$$ $$6795034624$$ $$[2, 2]$$ $$5120$$ $$0.58751$$
10304.r3 10304w1 $$[0, 0, 0, -236, 1360]$$ $$5545233/161$$ $$42205184$$ $$$$ $$2560$$ $$0.24094$$ $$\Gamma_0(N)$$-optimal
10304.r4 10304w4 $$[0, 0, 0, 1684, -21936]$$ $$2014698447/1958887$$ $$-513510473728$$ $$$$ $$10240$$ $$0.93409$$

## Rank

sage: E.rank()

The elliptic curves in class 10304.r have rank $$1$$.

## Complex multiplication

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

## Modular form 10304.2.a.r

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. 