# Properties

 Label 12100f Number of curves 4 Conductor 12100 CM no Rank 1 Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("12100.j1")

sage: E.isogeny_class()

## Elliptic curves in class 12100f

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
12100.j3 12100f1 [0, -1, 0, -4033, -66438] [2] 17280 $$\Gamma_0(N)$$-optimal
12100.j4 12100f2 [0, -1, 0, 11092, -459688] [2] 34560
12100.j1 12100f3 [0, -1, 0, -125033, 17055062] [2] 51840
12100.j2 12100f4 [0, -1, 0, -109908, 21320312] [2] 103680

## Rank

sage: E.rank()

The elliptic curves in class 12100f have rank $$1$$.

## Modular form 12100.2.a.j

sage: E.q_eigenform(10)

$$q + 2q^{3} + 2q^{7} + q^{9} + 2q^{13} - 6q^{17} + 4q^{19} + O(q^{20})$$

## 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 Cremona numbering.

$$\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels.