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Properties

Label 378.d
Number of curves $3$
Conductor $378$
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 378.d have rank \(0\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1 + T\)
\(3\)\(1\)
\(7\)\(1 - T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(5\) \( 1 + 5 T^{2}\) 1.5.a
\(11\) \( 1 + 11 T^{2}\) 1.11.a
\(13\) \( 1 - 5 T + 13 T^{2}\) 1.13.af
\(17\) \( 1 + 3 T + 17 T^{2}\) 1.17.d
\(19\) \( 1 - 2 T + 19 T^{2}\) 1.19.ac
\(23\) \( 1 - 9 T + 23 T^{2}\) 1.23.aj
\(29\) \( 1 - 3 T + 29 T^{2}\) 1.29.ad
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 378.d do not have complex multiplication.

Modular form 378.2.a.d

Copy content sage:E.q_eigenform(10)
 
\(q - q^{2} + q^{4} + q^{7} - q^{8} + 5 q^{13} - q^{14} + q^{16} - 3 q^{17} + 2 q^{19} + O(q^{20})\) 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 LMFDB numbering.

\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 378.d

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
378.d1 378b3 \([1, -1, 0, -1062, 13590]\) \(-545407363875/14\) \(-3402\) \([3]\) \(108\) \(0.19388\)  
378.d2 378b1 \([1, -1, 0, -12, 24]\) \(-7414875/2744\) \(-74088\) \([3]\) \(36\) \(-0.35542\) \(\Gamma_0(N)\)-optimal
378.d3 378b2 \([1, -1, 0, 93, -235]\) \(4492125/3584\) \(-70543872\) \([]\) \(108\) \(0.19388\)