# Properties

 Label 75106.a Number of curves 4 Conductor 75106 CM no Rank 1 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("75106.a1")

sage: E.isogeny_class()

## Elliptic curves in class 75106.a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
75106.a1 75106b4 [1, 0, 0, -249663, 33159175]  1271808
75106.a2 75106b3 [1, 0, 0, -227573, 41761021]  635904
75106.a3 75106b2 [1, 0, 0, -95033, -11281487]  423936
75106.a4 75106b1 [1, 0, 0, -6673, -130455]  211968 $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 75106.a have rank $$1$$.

## Modular form 75106.2.a.a

sage: E.q_eigenform(10)

$$q + q^{2} - 2q^{3} + q^{4} - 2q^{6} - 4q^{7} + q^{8} + q^{9} - 6q^{11} - 2q^{12} - 2q^{13} - 4q^{14} + q^{16} - q^{17} + q^{18} + 4q^{19} + O(q^{20})$$

## 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 LMFDB numbering.

$$\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 