# Properties

 Label 212415.w Number of curves 4 Conductor 212415 CM no Rank 1 Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("212415.w1")

sage: E.isogeny_class()

## Elliptic curves in class 212415.w

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
212415.w1 212415t4 [1, 0, 0, -23875741, -44903895730] [2] 14155776
212415.w2 212415t3 [1, 0, 0, -7590591, 7482033180] [2] 14155776
212415.w3 212415t2 [1, 0, 0, -1572166, -622377925] [2, 2] 7077888
212415.w4 212415t1 [1, 0, 0, 197959, -57000000] [2] 3538944 $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 212415.w have rank $$1$$.

## Modular form 212415.2.a.w

sage: E.q_eigenform(10)

$$q - q^{2} + q^{3} - q^{4} - q^{5} - q^{6} + 3q^{8} + q^{9} + q^{10} - q^{12} + 6q^{13} - q^{15} - q^{16} - q^{18} - 4q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels.