# Properties

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

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("12100.i1")

sage: E.isogeny_class()

## Elliptic curves in class 12100.i

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
12100.i1 12100g4 [0, -1, 0, -21478508, 38320869512] [2] 622080
12100.i2 12100g3 [0, -1, 0, -1347133, 594672762] [2] 311040
12100.i3 12100g2 [0, -1, 0, -303508, 36469512] [2] 207360
12100.i4 12100g1 [0, -1, 0, -137133, -19099738] [2] 103680 $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

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

## Modular form 12100.2.a.i

sage: E.q_eigenform(10)

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

## 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 LMFDB 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 LMFDB labels.