# Properties

 Label 80275a Number of curves $3$ Conductor $80275$ CM no Rank $1$ Graph

# Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 80275a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
80275.m3 80275a1 $$[0, -1, 1, 2817, -3482]$$ $$32768/19$$ $$-1432958921875$$ $$[]$$ $$77760$$ $$1.0220$$ $$\Gamma_0(N)$$-optimal
80275.m2 80275a2 $$[0, -1, 1, -39433, -3193357]$$ $$-89915392/6859$$ $$-517298170796875$$ $$[]$$ $$233280$$ $$1.5713$$
80275.m1 80275a3 $$[0, -1, 1, -3250433, -2254505732]$$ $$-50357871050752/19$$ $$-1432958921875$$ $$[]$$ $$699840$$ $$2.1206$$

## Rank

sage: E.rank()

The elliptic curves in class 80275a have rank $$1$$.

## Complex multiplication

The elliptic curves in class 80275a do not have complex multiplication.

## Modular form 80275.2.a.a

sage: E.q_eigenform(10)

$$q + 2q^{3} - 2q^{4} - q^{7} + q^{9} - 3q^{11} - 4q^{12} + 4q^{16} + 3q^{17} - q^{19} + O(q^{20})$$

## 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 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.