# Properties

 Label 3025a Number of curves $2$ Conductor $3025$ CM $$\Q(\sqrt{-11})$$ Rank $1$ Graph # Learn more about

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

sage: E.isogeny_class()

## Elliptic curves in class 3025a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality CM discriminant
3025.d2 3025a1 $$[0, 1, 1, -183, 919]$$ $$-32768$$ $$-20796875$$ $$[]$$ $$432$$ $$0.18165$$ $$\Gamma_0(N)$$-optimal $$-11$$
3025.d1 3025a2 $$[0, 1, 1, -22183, -1312206]$$ $$-32768$$ $$-36842932671875$$ $$[]$$ $$4752$$ $$1.3806$$   $$-11$$

## Rank

sage: E.rank()

The elliptic curves in class 3025a have rank $$1$$.

## Complex multiplication

Each elliptic curve in class 3025a has complex multiplication by an order in the imaginary quadratic field $$\Q(\sqrt{-11})$$.

## Modular form3025.2.a.a

sage: E.q_eigenform(10)

$$q + q^{3} - 2q^{4} - 2q^{9} - 2q^{12} + 4q^{16} + O(q^{20})$$ ## 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.

$$\left(\begin{array}{rr} 1 & 11 \\ 11 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels. 