# Properties

 Label 75a Number of curves $2$ Conductor $75$ CM no Rank $0$ Graph # Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 75a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
75.c1 75a1 $$[0, -1, 1, -8, -7]$$ $$-102400/3$$ $$-1875$$ $$[]$$ $$6$$ $$-0.59310$$ $$\Gamma_0(N)$$-optimal
75.c2 75a2 $$[0, -1, 1, 42, 443]$$ $$20480/243$$ $$-94921875$$ $$[]$$ $$30$$ $$0.21162$$

## Rank

sage: E.rank()

The elliptic curves in class 75a have rank $$0$$.

## Complex multiplication

The elliptic curves in class 75a do not have complex multiplication.

## Modular form75.2.a.a

sage: E.q_eigenform(10)

$$q + 2q^{2} - q^{3} + 2q^{4} - 2q^{6} - 3q^{7} + q^{9} + 2q^{11} - 2q^{12} + q^{13} - 6q^{14} - 4q^{16} + 2q^{17} + 2q^{18} - 5q^{19} + O(q^{20})$$ ## 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.

$$\left(\begin{array}{rr} 1 & 5 \\ 5 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels. 