Elliptic curve isogeny class with LMFDB label 78400.fp (Cremona label 78400gt)

• Label: 78400.fp
• Number of curves: $4$
• Conductor: $78400$
• CM: \(\Q(\sqrt{-1}) \)
• Rank: $1$

Rank

The elliptic curves in class 78400.fp have rank \(1\).

L-function data

Bad L-factors:

Prime	L-Factor
\(2\)	\(1\)
\(5\)	\(1\)
\(7\)	\(1\)

Good L-factors:

Prime	L-Factor	Isogeny Class over \(\mathbb{F}_p\)
\(3\)	\( 1 + 3 T^{2}\)	1.3.a
\(11\)	\( 1 + 11 T^{2}\)	1.11.a
\(13\)	\( 1 + 6 T + 13 T^{2}\)	1.13.g
\(17\)	\( 1 - 2 T + 17 T^{2}\)	1.17.ac
\(19\)	\( 1 + 19 T^{2}\)	1.19.a
\(23\)	\( 1 + 23 T^{2}\)	1.23.a
\(29\)	\( 1 - 10 T + 29 T^{2}\)	1.29.ak
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Complex multiplication

Each elliptic curve in class 78400.fp has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-1}) \).

Modular form 78400.2.a.fp

Isogeny matrix

The \((i,j)\)-th 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.

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

Isogeny graph

The vertices are labelled with LMFDB labels, and the \( \Gamma_0(N) \)-optimal curve is highlighted in blue.

Elliptic curves in class 78400.fp

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality	CM discriminant
78400.fp1	78400gt4	\([0, 0, 0, -53900, -4802000]\)	\(287496\)	\(60236288000000\)	\([2]\)	\(196608\)	\(1.5069\)		\(-16\)
78400.fp2	78400gt3	\([0, 0, 0, -53900, 4802000]\)	\(287496\)	\(60236288000000\)	\([2]\)	\(196608\)	\(1.5069\)		\(-16\)
78400.fp3	78400gt2	\([0, 0, 0, -4900, 0]\)	\(1728\)	\(7529536000000\)	\([2, 2]\)	\(98304\)	\(1.1603\)		\(-4\)
78400.fp4	78400gt1	\([0, 0, 0, 1225, 0]\)	\(1728\)	\(-117649000000\)	\([2]\)	\(49152\)	\(0.81371\)	\(\Gamma_0(N)\)-optimal	\(-4\)