# Properties

 Label 312.f Number of curves 4 Conductor 312 CM no Rank 0 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("312.f1")

sage: E.isogeny_class()

## Elliptic curves in class 312.f

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
312.f1 312c3 [0, 1, 0, -832, -9520]  64
312.f2 312c2 [0, 1, 0, -52, -160] [2, 2] 32
312.f3 312c1 [0, 1, 0, -7, 2]  16 $$\Gamma_0(N)$$-optimal
312.f4 312c4 [0, 1, 0, 8, -448]  64

## Rank

sage: E.rank()

The elliptic curves in class 312.f have rank $$0$$.

## Modular form312.2.a.f

sage: E.q_eigenform(10)

$$q + q^{3} + 2q^{5} + q^{9} + q^{13} + 2q^{15} + 2q^{17} - 4q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 