Properties

Label 2205.i
Number of curves $8$
Conductor $2205$
CM no
Rank $0$
Graph

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

Elliptic curves in class 2205.i

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
2205.i1 2205j7 \([1, -1, 0, -952569, -357605172]\) \(1114544804970241/405\) \(34735279005\) \([2]\) \(12288\) \(1.8131\)  
2205.i2 2205j5 \([1, -1, 0, -59544, -5574717]\) \(272223782641/164025\) \(14067787997025\) \([2, 2]\) \(6144\) \(1.4666\)  
2205.i3 2205j8 \([1, -1, 0, -48519, -7711362]\) \(-147281603041/215233605\) \(-18459751409696205\) \([2]\) \(12288\) \(1.8131\)  
2205.i4 2205j4 \([1, -1, 0, -35289, 2560410]\) \(56667352321/15\) \(1286491815\) \([2]\) \(3072\) \(1.1200\)  
2205.i5 2205j3 \([1, -1, 0, -4419, -51192]\) \(111284641/50625\) \(4341909875625\) \([2, 2]\) \(3072\) \(1.1200\)  
2205.i6 2205j2 \([1, -1, 0, -2214, 40095]\) \(13997521/225\) \(19297377225\) \([2, 2]\) \(1536\) \(0.77341\)  
2205.i7 2205j1 \([1, -1, 0, -9, 1728]\) \(-1/15\) \(-1286491815\) \([2]\) \(768\) \(0.42684\) \(\Gamma_0(N)\)-optimal
2205.i8 2205j6 \([1, -1, 0, 15426, -396495]\) \(4733169839/3515625\) \(-301521519140625\) \([2]\) \(6144\) \(1.4666\)  

Rank

sage: E.rank()
 

The elliptic curves in class 2205.i have rank \(0\).

Complex multiplication

The elliptic curves in class 2205.i do not have complex multiplication.

Modular form 2205.2.a.i

sage: E.q_eigenform(10)
 
\(q + q^{2} - q^{4} + q^{5} - 3q^{8} + q^{10} + 4q^{11} + 2q^{13} - q^{16} + 2q^{17} - 4q^{19} + O(q^{20})\)  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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.