# Properties

 Label 2310f Number of curves $4$ Conductor $2310$ CM no Rank $1$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("f1")

sage: E.isogeny_class()

## Elliptic curves in class 2310f

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
2310.f4 2310f1 $$[1, 0, 1, 256, 2126]$$ $$1865864036231/2993760000$$ $$-2993760000$$ $$$$ $$1280$$ $$0.50348$$ $$\Gamma_0(N)$$-optimal
2310.f3 2310f2 $$[1, 0, 1, -1744, 21326]$$ $$586145095611769/140040608400$$ $$140040608400$$ $$[2, 2]$$ $$2560$$ $$0.85005$$
2310.f2 2310f3 $$[1, 0, 1, -9444, -335954]$$ $$93137706732176569/5369647977540$$ $$5369647977540$$ $$$$ $$5120$$ $$1.1966$$
2310.f1 2310f4 $$[1, 0, 1, -26044, 1615406]$$ $$1953542217204454969/170843779260$$ $$170843779260$$ $$$$ $$5120$$ $$1.1966$$

## Rank

sage: E.rank()

The elliptic curves in class 2310f have rank $$1$$.

## Complex multiplication

The elliptic curves in class 2310f do not have complex multiplication.

## Modular form2310.2.a.f

sage: E.q_eigenform(10)

$$q - q^{2} + q^{3} + q^{4} - q^{5} - q^{6} - q^{7} - q^{8} + q^{9} + q^{10} + q^{11} + q^{12} - 2 q^{13} + q^{14} - q^{15} + q^{16} - 2 q^{17} - q^{18} + 4 q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels. 