# Properties

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

# Related objects

Show commands: SageMath
E = EllipticCurve("b1")

E.isogeny_class()

## Elliptic curves in class 1305.b

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
1305.b1 1305d3 $$[1, -1, 1, -2318, 42896]$$ $$1888690601881/31827645$$ $$23202353205$$ $$[2]$$ $$1536$$ $$0.78696$$
1305.b2 1305d2 $$[1, -1, 1, -293, -844]$$ $$3803721481/1703025$$ $$1241505225$$ $$[2, 2]$$ $$768$$ $$0.44038$$
1305.b3 1305d1 $$[1, -1, 1, -248, -1438]$$ $$2305199161/1305$$ $$951345$$ $$[2]$$ $$384$$ $$0.093810$$ $$\Gamma_0(N)$$-optimal
1305.b4 1305d4 $$[1, -1, 1, 1012, -7108]$$ $$157376536199/118918125$$ $$-86691313125$$ $$[2]$$ $$1536$$ $$0.78696$$

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form1305.2.a.b

sage: E.q_eigenform(10)

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

## 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 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.