# Properties

 Label 1305.e Number of curves $4$ Conductor $1305$ CM no Rank $0$ Graph

# Related objects

Show commands: SageMath
E = EllipticCurve("e1")

E.isogeny_class()

## Elliptic curves in class 1305.e

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
1305.e1 1305c3 $$[1, -1, 0, -62640, 6049971]$$ $$37286818682653441/1305$$ $$951345$$ $$[2]$$ $$2560$$ $$1.0934$$
1305.e2 1305c2 $$[1, -1, 0, -3915, 95256]$$ $$9104453457841/1703025$$ $$1241505225$$ $$[2, 2]$$ $$1280$$ $$0.74685$$
1305.e3 1305c4 $$[1, -1, 0, -3510, 115425]$$ $$-6561258219361/3978455625$$ $$-2900294150625$$ $$[2]$$ $$2560$$ $$1.0934$$
1305.e4 1305c1 $$[1, -1, 0, -270, 1215]$$ $$2992209121/951345$$ $$693530505$$ $$[2]$$ $$640$$ $$0.40028$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form1305.2.a.e

sage: E.q_eigenform(10)

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

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.