# Properties

 Label 77315f Number of curves $3$ Conductor $77315$ CM no Rank $0$ Graph

# Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 77315f

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
77315.d2 77315f1 $$[0, 1, 1, -2945, -69236]$$ $$-262144/35$$ $$-377272536515$$ $$[]$$ $$68172$$ $$0.95392$$ $$\Gamma_0(N)$$-optimal
77315.d3 77315f2 $$[0, 1, 1, 19145, 180381]$$ $$71991296/42875$$ $$-462158857230875$$ $$[]$$ $$204516$$ $$1.5032$$
77315.d1 77315f3 $$[0, 1, 1, -290115, 62820994]$$ $$-250523582464/13671875$$ $$-147372084576171875$$ $$[]$$ $$613548$$ $$2.0525$$

## Rank

sage: E.rank()

The elliptic curves in class 77315f have rank $$0$$.

## Complex multiplication

The elliptic curves in class 77315f do not have complex multiplication.

## Modular form 77315.2.a.f

sage: E.q_eigenform(10)

$$q + q^{3} - 2q^{4} + q^{5} + q^{7} - 2q^{9} + 3q^{11} - 2q^{12} - 5q^{13} + q^{15} + 4q^{16} + 3q^{17} - 2q^{19} + O(q^{20})$$

## 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.