Elliptic curve with LMFDB label 10830.bb1 (Cremona label 10830ba2)

• Label: 10830.bb1
• Conductor: $10830$
• Discriminant: $-4.582\times 10^{12}$
• j-invariant: \( -\frac{27692833539889}{35156250} \)
• CM: no
• Rank: $1$
• Torsion structure: trivial

Minimal Weierstrass equation

\( y^2 + x y = x^{3} - 31956 x - 2203830 \)

Mordell-Weil group structure

\(\Z\)

Infinite order Mordell-Weil generator and height

\(P\)	=	\( \left(\frac{1155}{4}, \frac{27315}{8}\right) \)
\(\hat{h}(P)\)	≈	$6.8084176723294$

Integral points

None

Invariants

Conductor:	\( 10830 \)	=	\(2 \cdot 3 \cdot 5 \cdot 19^{2}\)
Discriminant:	\(-4581597656250 \)	=	\(-1 \cdot 2 \cdot 3^{2} \cdot 5^{9} \cdot 19^{4} \)
j-invariant:	\( -\frac{27692833539889}{35156250} \)	=	\(-1 \cdot 2^{-1} \cdot 3^{-2} \cdot 5^{-9} \cdot 7^{3} \cdot 19^{2} \cdot 607^{3}\)
Endomorphism ring:	\(\Z\)
Geometric endomorphism ring:	\(\Z\)	(no potential complex multiplication)
Sato-Tate group:	$\mathrm{SU}(2)$

BSD invariants

Rank:	\(1\)
Regulator:	\(6.80841767233\)
Real period:	\(0.178524707943\)
Tamagawa product:	\( 6 \)  = \( 1\cdot2\cdot1\cdot3 \)
Torsion order:	\(1\)
Analytic order of Ш:	\(1\) (exact)

Modular invariants

Modular form 10830.2.a.bb

\( q + q^{2} + q^{3} + q^{4} - q^{5} + q^{6} - q^{7} + q^{8} + q^{9} - q^{10} - 3q^{11} + q^{12} + 2q^{13} - q^{14} - q^{15} + q^{16} - 6q^{17} + q^{18} + O(q^{20}) \)

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

Modular degree:	38880
\( \Gamma_0(N) \)-optimal:	no
Manin constant:	1

Special L-value

\( L'(E,1) \) ≈ \( 7.29282465903 \)

Local data

This elliptic curve is not semistable.

prime	Tamagawa number	Kodaira symbol	Reduction type	Root number	ord(\(N\))	ord(\(\Delta\))	ord\((j)_{-}\)
\(2\)	\(1\)	\( I_{1} \)	Split multiplicative	-1	1	1	1
\(3\)	\(2\)	\( I_{2} \)	Split multiplicative	-1	1	2	2
\(5\)	\(1\)	\( I_{9} \)	Non-split multiplicative	1	1	9	9
\(19\)	\(3\)	\( IV \)	Additive	1	2	4	0

Galois representations

The 2-adic representation attached to this elliptic curve is surjective.

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

prime	Image of Galois representation
\(3\)	B.1.2

$p$-adic data

$p$-adic regulators

\(p\)-adic regulators are not yet computed for curves that are not \(\Gamma_0\)-optimal.

Iwasawa invariants

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47
Reduction type	split	split	nonsplit	ordinary	ordinary	ordinary	ordinary	add	ordinary	ss	ordinary	ordinary	ordinary	ordinary	ss
$\lambda$-invariant(s)	2	4	1	1	1	1	1	-	1	1,1	1	1	1	1	1,1
$\mu$-invariant(s)	0	1	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 \(d=\) 3.
Its isogeny class 10830.bb consists of 2 curves linked by isogenies of degree 3.

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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:

$[K:\Q]$	$K$	$E(K)_{\rm tors}$	Base change curve
$2$	\(\Q(\sqrt{-3}) \)	\(\Z/3\Z\)	Not in database
$3$	3.1.14440.1	\(\Z/2\Z\)	Not in database
$3$	3.1.350892.3	\(\Z/3\Z\)	Not in database
$6$	6.0.5629867200.9	\(\Z/6\Z\)	Not in database
$6$	6.0.8340544000.1	\(\Z/2\Z \times \Z/2\Z\)	Not in database
$6$	6.0.369375586992.2	\(\Z/3\Z \times \Z/3\Z\)	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.