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Properties

Label 1230.f
Number of curves $8$
Conductor $1230$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("f1")
 
E.isogeny_class()
 

Elliptic curves in class 1230.f

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
1230.f1 1230f7 \([1, 1, 1, -14346720, -20921901465]\) \(326573981641149886485204481/2690010\) \(2690010\) \([2]\) \(24576\) \(2.2516\)  
1230.f2 1230f5 \([1, 1, 1, -896670, -327184905]\) \(79729981196639723693281/7236153800100\) \(7236153800100\) \([2, 2]\) \(12288\) \(1.9051\)  
1230.f3 1230f8 \([1, 1, 1, -894620, -328752745]\) \(-79184385609230668294081/759738277429254810\) \(-759738277429254810\) \([2]\) \(24576\) \(2.2516\)  
1230.f4 1230f3 \([1, 1, 1, -56170, -5105305]\) \(19599160390581221281/185398179210000\) \(185398179210000\) \([2, 4]\) \(6144\) \(1.5585\)  
1230.f5 1230f6 \([1, 1, 1, -15670, -12265705]\) \(-425532204913949281/64677894355880100\) \(-64677894355880100\) \([4]\) \(12288\) \(1.9051\)  
1230.f6 1230f2 \([1, 1, 1, -6170, 54695]\) \(25976677550021281/13616100000000\) \(13616100000000\) \([2, 4]\) \(3072\) \(1.2119\)  
1230.f7 1230f1 \([1, 1, 1, -4890, 129447]\) \(12931706531187361/15114240000\) \(15114240000\) \([8]\) \(1536\) \(0.86534\) \(\Gamma_0(N)\)-optimal
1230.f8 1230f4 \([1, 1, 1, 23350, 456167]\) \(1407936942337442399/900878906250000\) \(-900878906250000\) \([4]\) \(6144\) \(1.5585\)  

Rank

sage: E.rank()
 

The elliptic curves in class 1230.f have rank \(0\).

Complex multiplication

The elliptic curves in class 1230.f do not have complex multiplication.

Modular form 1230.2.a.f

sage: E.q_eigenform(10)
 
\(q + q^{2} - q^{3} + q^{4} + q^{5} - q^{6} + q^{8} + q^{9} + q^{10} + 4 q^{11} - q^{12} - 2 q^{13} - q^{15} + q^{16} + 2 q^{17} + q^{18} - 4 q^{19} + O(q^{20})\) Copy content Toggle raw display

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.

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 8 & 16 & 16 \\ 2 & 1 & 2 & 2 & 4 & 4 & 8 & 8 \\ 4 & 2 & 1 & 4 & 8 & 8 & 16 & 16 \\ 4 & 2 & 4 & 1 & 2 & 2 & 4 & 4 \\ 8 & 4 & 8 & 2 & 1 & 4 & 8 & 8 \\ 8 & 4 & 8 & 2 & 4 & 1 & 2 & 2 \\ 16 & 8 & 16 & 4 & 8 & 2 & 1 & 4 \\ 16 & 8 & 16 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.