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Properties

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

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Show commands: SageMath
Copy content sage:E = EllipticCurve([1, -1, 0, -9, 1728]) E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

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

L-function data

 
Bad L-factors:
Prime L-Factor
\(3\)\(1\)
\(5\)\(1 + T\)
\(7\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(2\) \( 1 - 2 T + 2 T^{2}\) 1.2.ac
\(11\) \( 1 + T + 11 T^{2}\) 1.11.b
\(13\) \( 1 + 3 T + 13 T^{2}\) 1.13.d
\(17\) \( 1 + 3 T + 17 T^{2}\) 1.17.d
\(19\) \( 1 + 6 T + 19 T^{2}\) 1.19.g
\(23\) \( 1 - 4 T + 23 T^{2}\) 1.23.ae
\(29\) \( 1 - T + 29 T^{2}\) 1.29.ab
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

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

Modular form 2205.2.a.j

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

Isogeny matrix

Copy content 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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 2205j

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