# Properties

 Label 12495h Number of curves 4 Conductor 12495 CM no Rank 0 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("12495.c1")

sage: E.isogeny_class()

## Elliptic curves in class 12495h

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
12495.c4 12495h1 [1, 1, 1, 685, -11320]  12288 $$\Gamma_0(N)$$-optimal
12495.c3 12495h2 [1, 1, 1, -5440, -128920] [2, 2] 24576
12495.c1 12495h3 [1, 1, 1, -82615, -9173830]  49152
12495.c2 12495h4 [1, 1, 1, -26265, 1512090]  49152

## Rank

sage: E.rank()

The elliptic curves in class 12495h have rank $$0$$.

## Modular form 12495.2.a.c

sage: E.q_eigenform(10)

$$q - q^{2} - q^{3} - q^{4} + q^{5} + q^{6} + 3q^{8} + q^{9} - q^{10} + q^{12} + 6q^{13} - q^{15} - q^{16} + q^{17} - q^{18} - 4q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels. 