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Properties

Label 271062d
Number of curves $4$
Conductor $271062$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("d1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 271062d have rank \(0\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1 + T\)
\(3\)\(1\)
\(11\)\(1 + T\)
\(37\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(5\) \( 1 + 2 T + 5 T^{2}\) 1.5.c
\(7\) \( 1 + 4 T + 7 T^{2}\) 1.7.e
\(13\) \( 1 + 6 T + 13 T^{2}\) 1.13.g
\(17\) \( 1 + 2 T + 17 T^{2}\) 1.17.c
\(19\) \( 1 + 19 T^{2}\) 1.19.a
\(23\) \( 1 + 8 T + 23 T^{2}\) 1.23.i
\(29\) \( 1 - 10 T + 29 T^{2}\) 1.29.ak
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 271062d do not have complex multiplication.

Modular form 271062.2.a.d

Copy content sage:E.q_eigenform(10)
 
\(q - q^{2} + q^{4} - 2 q^{5} - 4 q^{7} - q^{8} + 2 q^{10} - q^{11} - 6 q^{13} + 4 q^{14} + q^{16} - 2 q^{17} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 271062d

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
271062.d4 271062d1 \([1, -1, 0, -280752723, -1561450284491]\) \(1308451928740468777/194033737531392\) \(362923526288903585358938112\) \([2]\) \(168099840\) \(3.8209\) \(\Gamma_0(N)\)-optimal
271062.d2 271062d2 \([1, -1, 0, -4318098003, -109212417361355]\) \(4760617885089919932457/133756441657344\) \(250179994921170002376720384\) \([2, 2]\) \(336199680\) \(4.1675\)  
271062.d3 271062d3 \([1, -1, 0, -4144618323, -118389665913035]\) \(-4209586785160189454377/801182513521564416\) \(-1498543432227661234168753502976\) \([2]\) \(672399360\) \(4.5141\)  
271062.d1 271062d4 \([1, -1, 0, -69089102163, -6989745509872331]\) \(19499096390516434897995817/15393430272\) \(28792095988424460417792\) \([2]\) \(672399360\) \(4.5141\)