Elliptic Curve with LMFDB label 15.a1 (Cremona label 15a5)

• Label: 15.a1
• Conductor: 15
• Discriminant: 405
• j-invariant: \( \frac{1114544804970241}{405} \)
• CM: no
• Rank: 0
• Torsion Structure: \(\Z/{2}\Z\)

Minimal Weierstrass equation

\( y^2 + x y + y = x^{3} + x^{2} - 2160 x - 39540 \)

Mordell-Weil group structure

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

Torsion generators

\( \left(-\frac{109}{4}, \frac{105}{8}\right) \)

Integral points

None

Invariants

Conductor:	\( 15 \)	=	\(3 \cdot 5\)
Discriminant:	\(405 \)	=	\(3^{4} \cdot 5 \)
j-invariant:	\( \frac{1114544804970241}{405} \)	=	\(3^{-4} \cdot 5^{-1} \cdot 103681^{3}\)
Endomorphism ring:	\(\Z\)		(no Complex Multiplication)
Sato-Tate Group:	$\mathrm{SU}(2)$

BSD invariants

Rank:	\(0\)
Regulator:	\(1\)
Real period:	\(0.700301521166\)
Tamagawa product:	\( 2 \)  = \( 2\cdot1 \)
Torsion order:	\(2\)
Analytic order of Ш:	\(1\) (exact)

Modular invariants

Modular form 15.2.a.a

\( q - q^{2} - q^{3} - q^{4} + q^{5} + q^{6} + 3q^{8} + q^{9} - q^{10} - 4q^{11} + q^{12} - 2q^{13} - q^{15} - q^{16} + 2q^{17} - q^{18} + 4q^{19} + O(q^{20}) \)

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

Modular degree:	4
\( \Gamma_0(N) \)-optimal:	no
Manin constant:	1

Special L-value

\( L(E,1) \) ≈ \( 0.350150760583 \)

Local data

prime	Tamagawa number	Kodaira symbol	Reduction type	Root number	ord(\(N\))	ord(\(\Delta\))	ord\((j)_{-}\)
\(3\)	\(2\)	\( I_{4} \)	Non-split multiplicative	1	1	4	4
\(5\)	\(1\)	\( I_{1} \)	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 X225g.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^4\Z_2)$ generated by $\left(\begin{array}{rr} 5 & 5 \\ 0 & 5 \end{array}\right),\left(\begin{array}{rr} 5 & 5 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 5 \\ 0 & 3 \end{array}\right)$ and has index 96.

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

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

Iwasawa invariants

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

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 3$ of good reduction are zero.

Isogenies

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 4, 8 and 16.
Its isogeny class 15.a consists of 8 curves linked by isogenies of degrees dividing 16.

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	\(\Q(\sqrt{-5}) \)	\(\Z/4\Z\)	Not in database
	\(\Q(\sqrt{-1}) \)	\(\Z/4\Z\)	2.0.4.1-225.2-a10
	\(\Q(\sqrt{5}) \)	\(\Z/2\Z \times \Z/2\Z\)	2.2.5.1-45.1-a9
4	\(\Q(\zeta_{8})\)	\(\Z/8\Z\)	Not in database
	\(\Q(i, \sqrt{10})\)	\(\Z/8\Z\)	Not in database
	4.0.32000.1	\(\Z/8\Z\)	Not in database
	\(\Q(i, \sqrt{5})\)	\(\Z/2\Z \times \Z/4\Z\)	Not in database
	4.2.2000.1	\(\Z/2\Z \times \Z/4\Z\)	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.