# Properties

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

# Related objects

Show commands: SageMath
E = EllipticCurve("d1")

E.isogeny_class()

## Elliptic curves in class 1305.d

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
1305.d1 1305f1 $$[0, 0, 1, -102, -405]$$ $$-160989184/3915$$ $$-2854035$$ $$[]$$ $$160$$ $$0.024133$$ $$\Gamma_0(N)$$-optimal
1305.d2 1305f2 $$[0, 0, 1, 438, -1728]$$ $$12747309056/9145875$$ $$-6667342875$$ $$[3]$$ $$480$$ $$0.57344$$

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form1305.2.a.d

sage: E.q_eigenform(10)

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

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.