# Properties

 Label 12696.g Number of curves $4$ Conductor $12696$ CM no Rank $1$ Graph

# Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 12696.g

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
12696.g1 12696o3 $$[0, -1, 0, -397984, -81644900]$$ $$45989074372/7555707$$ $$1145360182034967552$$ $$[2]$$ $$202752$$ $$2.1868$$
12696.g2 12696o2 $$[0, -1, 0, -112324, 13308484]$$ $$4135597648/385641$$ $$14614709317081344$$ $$[2, 2]$$ $$101376$$ $$1.8402$$
12696.g3 12696o1 $$[0, -1, 0, -109679, 14017344]$$ $$61604313088/621$$ $$1470884593104$$ $$[4]$$ $$50688$$ $$1.4937$$ $$\Gamma_0(N)$$-optimal
12696.g4 12696o4 $$[0, -1, 0, 131016, 62852508]$$ $$1640689628/12223143$$ $$-1852890972548226048$$ $$[2]$$ $$202752$$ $$2.1868$$

## Rank

sage: E.rank()

The elliptic curves in class 12696.g have rank $$1$$.

## Complex multiplication

The elliptic curves in class 12696.g do not have complex multiplication.

## Modular form 12696.2.a.g

sage: E.q_eigenform(10)

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

## 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.