# Properties

 Label 10115g Number of curves $3$ Conductor $10115$ CM no Rank $0$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("g1")

sage: E.isogeny_class()

## Elliptic curves in class 10115g

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
10115.f2 10115g1 $$[0, -1, 1, -385, 3358]$$ $$-262144/35$$ $$-844814915$$ $$[]$$ $$3360$$ $$0.44546$$ $$\Gamma_0(N)$$-optimal
10115.f3 10115g2 $$[0, -1, 1, 2505, -9069]$$ $$71991296/42875$$ $$-1034898270875$$ $$[]$$ $$10080$$ $$0.99476$$
10115.f1 10115g3 $$[0, -1, 1, -37955, -2964672]$$ $$-250523582464/13671875$$ $$-330005826171875$$ $$[]$$ $$30240$$ $$1.5441$$

## Rank

sage: E.rank()

The elliptic curves in class 10115g have rank $$0$$.

## Complex multiplication

The elliptic curves in class 10115g do not have complex multiplication.

## Modular form 10115.2.a.g

sage: E.q_eigenform(10)

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