# Properties

 Label 50575m Number of curves $3$ Conductor $50575$ CM no Rank $0$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("m1")

sage: E.isogeny_class()

## Elliptic curves in class 50575m

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
50575.t2 50575m1 $$[0, 1, 1, -9633, 400519]$$ $$-262144/35$$ $$-13200233046875$$ $$[]$$ $$80640$$ $$1.2502$$ $$\Gamma_0(N)$$-optimal
50575.t3 50575m2 $$[0, 1, 1, 62617, -1008356]$$ $$71991296/42875$$ $$-16170285482421875$$ $$[]$$ $$241920$$ $$1.7995$$
50575.t1 50575m3 $$[0, 1, 1, -948883, -372481731]$$ $$-250523582464/13671875$$ $$-5156341033935546875$$ $$[]$$ $$725760$$ $$2.3488$$

## Rank

sage: E.rank()

The elliptic curves in class 50575m have rank $$0$$.

## Complex multiplication

The elliptic curves in class 50575m do not have complex multiplication.

## Modular form 50575.2.a.m

sage: E.q_eigenform(10)

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