# Properties

 Label 1305.f Number of curves $2$ Conductor $1305$ CM no Rank $0$ Graph

# Related objects

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

E.isogeny_class()

## Elliptic curves in class 1305.f

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
1305.f1 1305g1 $$[1, -1, 0, -24, -37]$$ $$2146689/145$$ $$105705$$ $$[2]$$ $$128$$ $$-0.28720$$ $$\Gamma_0(N)$$-optimal
1305.f2 1305g2 $$[1, -1, 0, 21, -190]$$ $$1367631/21025$$ $$-15327225$$ $$[2]$$ $$256$$ $$0.059378$$

## Rank

sage: E.rank()

The elliptic curves in class 1305.f have rank $$0$$.

## Complex multiplication

The elliptic curves in class 1305.f do not have complex multiplication.

## Modular form1305.2.a.f

sage: E.q_eigenform(10)

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

## 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 LMFDB numbering.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.