# Properties

 Label 77315d Number of curves $2$ Conductor $77315$ CM no Rank $1$ Graph

# Related objects

Show commands: SageMath
sage: E = EllipticCurve("d1")

sage: E.isogeny_class()

## Elliptic curves in class 77315d

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
77315.f1 77315d1 $$[1, -1, 0, -11459, 135888]$$ $$15438249/8225$$ $$88659046081025$$ $$[2]$$ $$194304$$ $$1.3677$$ $$\Gamma_0(N)$$-optimal
77315.f2 77315d2 $$[1, -1, 0, 43766, 1030533]$$ $$860085351/541205$$ $$-5833765232131445$$ $$[2]$$ $$388608$$ $$1.7142$$

## Rank

sage: E.rank()

The elliptic curves in class 77315d have rank $$1$$.

## Complex multiplication

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

## Modular form 77315.2.a.d

sage: E.q_eigenform(10)

$$q + q^{2} - q^{4} + q^{5} - q^{7} - 3q^{8} - 3q^{9} + q^{10} - 6q^{11} - 2q^{13} - q^{14} - q^{16} - 4q^{17} - 3q^{18} - 4q^{19} + O(q^{20})$$

## 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 Cremona numbering.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels.