Elliptic Curve with LMFDB label 75150.m1 (Cremona label 75150r1)

• Label: 75150.m1
• Conductor: 75150
• Discriminant: 11413406250000
• j-invariant: \( \frac{83453453}{8016} \)
• CM: no
• Rank: 1
• Torsion Structure: \(\Z/{2}\Z\)

Minimal Weierstrass equation

\( y^2 + x y = x^{3} - x^{2} - 10242 x + 366916 \)

Mordell-Weil group structure

\(\Z\times \Z/{2}\Z\)

Infinite order Mordell-Weil generator and height

\(P\)	=	\( \left(-81, 853\right) \)
\(\hat{h}(P)\)	≈	2.61710979018

Torsion generators

\( \left(44, -22\right) \)

Integral points

\( \left(-81, 853\right) \), \( \left(44, -22\right) \), \( \left(80, 194\right) \)

Note: only one of each pair $\pm P$ is listed.

Invariants

Conductor:	\( 75150 \)	=	\(2 \cdot 3^{2} \cdot 5^{2} \cdot 167\)
Discriminant:	\(11413406250000 \)	=	\(2^{4} \cdot 3^{7} \cdot 5^{9} \cdot 167 \)
j-invariant:	\( \frac{83453453}{8016} \)	=	\(2^{-4} \cdot 3^{-1} \cdot 19^{3} \cdot 23^{3} \cdot 167^{-1}\)
Endomorphism ring:	\(\Z\)		(no Complex Multiplication)
Sato-Tate Group:	$\mathrm{SU}(2)$

BSD invariants

Rank:	\(1\)
Regulator:	\(2.61710979018\)
Real period:	\(0.697214132865\)
Tamagawa product:	\( 8 \)  = \( 2\cdot2\cdot2\cdot1 \)
Torsion order:	\(2\)
Analytic order of Ш:	\(1\) (exact)

Modular invariants

Modular form 75150.2.a.m

\( q - q^{2} + q^{4} - q^{8} - 2q^{11} - 6q^{13} + q^{16} + 4q^{17} - 8q^{19} + O(q^{20}) \)

For more coefficients, see the Downloads section to the right.

Modular degree:	230400
\( \Gamma_0(N) \)-optimal:	yes
Manin constant:	1

Special L-value

\( L'(E,1) \) ≈ \( 3.64937186595 \)

Local data

prime	Tamagawa number	Kodaira symbol	Reduction type	Root number	ord(\(N\))	ord(\(\Delta\))	ord\((j)_{-}\)
\(2\)	\(2\)	\( I_{4} \)	Non-split multiplicative	1	1	4	4
\(3\)	\(2\)	\( I_1^{*} \)	Additive	-1	2	7	1
\(5\)	\(2\)	\( III^{*} \)	Additive	-1	2	9	0
\(167\)	\(1\)	\( I_{1} \)	Non-split multiplicative	1	1	1	1

Galois representations

The image of the 2-adic representation attached to this elliptic curve is the subgroup of $\GL(2,\Z_2)$ with Rouse label X6.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^1\Z_2)$ generated by $\left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right)$ and has index 3.

The mod \( p \) Galois representation has maximal image \(\GL(2,\F_p)\) for all primes \( p \) except those listed.

prime	Image of Galois representation
\(2\)	B

$p$-adic data

$p$-adic regulators

Note: \(p\)-adic regulator data only exists for primes \(p\ge5\) of good ordinary reduction.

Iwasawa invariants

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	167
Reduction type	nonsplit	add	add	ss	ordinary	ordinary	ordinary	ordinary	ss	ordinary	ordinary	ordinary	ordinary	ordinary	ordinary	nonsplit
$\lambda$-invariant(s)	2	-	-	3,1	1	1	1	1	1,1	1	1	1	1	1	1	1
$\mu$-invariant(s)	0	-	-	0,0	0	0	0	0	0,0	0	0	0	0	0	0	0

An entry - indicates that the invariants are not computed because the reduction is additive.

Isogenies

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2.
Its isogeny class 75150.m consists of 2 curves linked by isogenies of degree 2.

Growth of torsion in number fields

The number fields $K$ of degree up to 7 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:

$[K:\Q]$	$K$	$E(K)_{\rm tors}$	Base-change curve
2	\(\Q(\sqrt{2505}) \)	\(\Z/2\Z \times \Z/2\Z\)	Not in database
4	4.0.1002000.4	\(\Z/4\Z\)	Not in database

We only show fields where the torsion growth is primitive. For each field $K$ we either show its label, or a defining polynomial when $K$ is not in the database.