# Properties

 Label 12635a Number of curves $3$ Conductor $12635$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("a1")

sage: E.isogeny_class()

## Elliptic curves in class 12635a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
12635.e2 12635a1 $$[0, -1, 1, -481, -4349]$$ $$-262144/35$$ $$-1646605835$$ $$[]$$ $$4752$$ $$0.50107$$ $$\Gamma_0(N)$$-optimal
12635.e3 12635a2 $$[0, -1, 1, 3129, 10452]$$ $$71991296/42875$$ $$-2017092147875$$ $$[]$$ $$14256$$ $$1.0504$$
12635.e1 12635a3 $$[0, -1, 1, -47411, 4172421]$$ $$-250523582464/13671875$$ $$-643205404296875$$ $$[]$$ $$42768$$ $$1.5997$$

## Rank

sage: E.rank()

The elliptic curves in class 12635a have rank $$1$$.

## Complex multiplication

The elliptic curves in class 12635a do not have complex multiplication.

## Modular form 12635.2.a.a

sage: E.q_eigenform(10)

$$q - q^{3} - 2q^{4} - q^{5} + q^{7} - 2q^{9} - 3q^{11} + 2q^{12} - 5q^{13} + q^{15} + 4q^{16} + 3q^{17} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels.