Elliptic Curve with LMFDB label 30.a7 (Cremona label 30a3)

• Label: 30.a7
• Conductor: 30
• Discriminant: -1536000
• j-invariant: \( -\frac{273359449}{1536000} \)
• CM: no
• Rank: 0
• Torsion Structure: \(\Z/{2}\Z\)

Minimal Weierstrass equation

\( y^2 + x y + y = x^{3} - 14 x - 64 \)

Mordell-Weil group structure

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

Torsion generators

\( \left(5, -3\right) \)

Integral points

\( \left(5, -3\right) \)

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

Invariants

Conductor:	\( 30 \)	=	\(2 \cdot 3 \cdot 5\)
Discriminant:	\(-1536000 \)	=	\(-1 \cdot 2^{12} \cdot 3 \cdot 5^{3} \)
j-invariant:	\( -\frac{273359449}{1536000} \)	=	\(-1 \cdot 2^{-12} \cdot 3^{-1} \cdot 5^{-3} \cdot 11^{3} \cdot 59^{3}\)
Endomorphism ring:	\(\Z\)		(no Complex Multiplication)
Sato-Tate Group:	$\mathrm{SU}(2)$

BSD invariants

Rank:	\(0\)
Regulator:	\(1\)
Real period:	\(1.11731608641\)
Tamagawa product:	\( 2 \)  = \( 2\cdot1\cdot1 \)
Torsion order:	\(2\)
Analytic order of Ш:	\(1\) (exact)

Modular invariants

Modular form 30.2.a.a

\( q - q^{2} + q^{3} + q^{4} - q^{5} - q^{6} - 4q^{7} - q^{8} + q^{9} + q^{10} + q^{12} + 2q^{13} + 4q^{14} - q^{15} + q^{16} + 6q^{17} - q^{18} - 4q^{19} + O(q^{20}) \)

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

Modular degree:	6
\( \Gamma_0(N) \)-optimal:	no
Manin constant:	1

Special L-value

\( L(E,1) \) ≈ \( 0.558658043207 \)

Local data

prime	Tamagawa number	Kodaira symbol	Reduction type	Root number	ord(\(N\))	ord(\(\Delta\))	ord\((j)_{-}\)
\(2\)	\(2\)	\( I_{12} \)	Non-split multiplicative	1	1	12	12
\(3\)	\(1\)	\( I_{1} \)	Split multiplicative	-1	1	1	1
\(5\)	\(1\)	\( I_{3} \)	Non-split multiplicative	1	1	3	3

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 X13c.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^3\Z_2)$ generated by $\left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 0 & 3 \end{array}\right),\left(\begin{array}{rr} 3 & 0 \\ 4 & 1 \end{array}\right)$ and has index 12.

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
\(3\)	B.1.2

$p$-adic data

$p$-adic regulators

All \(p\)-adic regulators are identically \(1\) since the rank is \(0\).

Iwasawa invariants

$p$	2	3	5
Reduction type	nonsplit	split	nonsplit
$\lambda$-invariant(s)	0	1	0
$\mu$-invariant(s)	0	1	0

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ of good reduction are zero.

Isogenies

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 3, 4, 6 and 12.
Its isogeny class 30.a consists of 8 curves linked by isogenies of degrees dividing 12.

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	\(\Q(\sqrt{5}) \)	\(\Z/4\Z\)	2.2.5.1-180.1-a1
	\(\Q(\sqrt{-15}) \)	\(\Z/2\Z \times \Z/2\Z\)	Not in database
	\(\Q(\sqrt{-3}) \)	\(\Z/12\Z\)	2.0.3.1-300.1-a1
3	3.1.243.1	\(\Z/6\Z\)	Not in database
4	\(\Q(\sqrt{-3}, \sqrt{5})\)	\(\Z/2\Z \times \Z/12\Z\)	Not in database
6	6.2.7381125.1	\(\Z/12\Z\)	Not in database
	6.0.22143375.1	\(\Z/2\Z \times \Z/6\Z\)	Not in database
	6.0.177147.2	\(\Z/3\Z \times \Z/12\Z\)	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.