Elliptic curve isogeny class with Cremona label 12225b (LMFDB label 12225.d)

• Label: 12225b
• Number of curves: $4$
• Conductor: $12225$
• CM: no
• Rank: $0$

Rank

The elliptic curves in class 12225b have rank \(0\).

L-function data

Bad L-factors:

Prime	L-Factor
\(3\)	\(1 - T\)
\(5\)	\(1\)
\(163\)	\(1 + T\)

Good L-factors:

Prime	L-Factor	Isogeny Class over \(\mathbb{F}_p\)
\(2\)	\( 1 + 2 T + 2 T^{2}\)	1.2.c
\(7\)	\( 1 + 7 T^{2}\)	1.7.a
\(11\)	\( 1 + 4 T + 11 T^{2}\)	1.11.e
\(13\)	\( 1 - 2 T + 13 T^{2}\)	1.13.ac
\(17\)	\( 1 - 4 T + 17 T^{2}\)	1.17.ae
\(19\)	\( 1 - 4 T + 19 T^{2}\)	1.19.ae
\(23\)	\( 1 - 6 T + 23 T^{2}\)	1.23.ag
\(29\)	\( 1 - 4 T + 29 T^{2}\)	1.29.ae
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Complex multiplication

The elliptic curves in class 12225b do not have complex multiplication.

Modular form 12225.2.a.b

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 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

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

Elliptic curves in class 12225b

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
12225.d3	12225b1	\([1, 1, 1, -20838, -1166094]\)	\(64043209720729/24755625\)	\(386806640625\)	\([2]\)	\(23040\)	\(1.1888\)	\(\Gamma_0(N)\)-optimal
12225.d2	12225b2	\([1, 1, 1, -23963, -797344]\)	\(97393143178729/39221822025\)	\(612840969140625\)	\([2, 2]\)	\(46080\)	\(1.5354\)
12225.d1	12225b3	\([1, 1, 1, -175838, 27755156]\)	\(38480618749557529/857682789615\)	\(13401293587734375\)	\([4]\)	\(92160\)	\(1.8820\)
12225.d4	12225b4	\([1, 1, 1, 77912, -5687344]\)	\(3347467708032071/2841729286815\)	\(-44402020106484375\)	\([2]\)	\(92160\)	\(1.8820\)