Properties

Label 374850la1
Conductor $374850$
Discriminant $-5.879\times 10^{18}$
j-invariant \( \frac{103823}{4386816} \)
CM no
Rank $2$
Torsion structure \(\Z/{2}\Z\)

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Minimal Weierstrass equation

sage: E = EllipticCurve([1, -1, 1, 10795, 116650797])
 
gp: E = ellinit([1, -1, 1, 10795, 116650797])
 
magma: E := EllipticCurve([1, -1, 1, 10795, 116650797]);
 

\(y^2+xy+y=x^3-x^2+10795x+116650797\)  Toggle raw display

Mordell-Weil group structure

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

Infinite order Mordell-Weil generators and heights

sage: E.gens()
 
magma: Generators(E);
 

\(P\) =  \(\left(1619, 65340\right)\)  Toggle raw display\(\left(149, 10950\right)\)  Toggle raw display
\(\hat{h}(P)\) ≈  $0.73433981628032745869324820095$$0.89654046942400522240009647280$

Torsion generators

sage: E.torsion_subgroup().gens()
 
gp: elltors(E)
 
magma: TorsionSubgroup(E);
 

\( \left(-481, 240\right) \)  Toggle raw display

Integral points

sage: E.integral_points()
 
magma: IntegralPoints(E);
 

\( \left(-481, 240\right) \), \( \left(-361, 8280\right) \), \( \left(-361, -7920\right) \), \( \left(-285, 9648\right) \), \( \left(-285, -9364\right) \), \( \left(-145, 10656\right) \), \( \left(-145, -10512\right) \), \( \left(-1, 10800\right) \), \( \left(-1, -10800\right) \), \( \left(3, 10800\right) \), \( \left(3, -10804\right) \), \( \left(149, 10950\right) \), \( \left(149, -11100\right) \), \( \left(303, 12000\right) \), \( \left(303, -12304\right) \), \( \left(419, 13740\right) \), \( \left(419, -14160\right) \), \( \left(639, 19280\right) \), \( \left(639, -19920\right) \), \( \left(989, 32580\right) \), \( \left(989, -33570\right) \), \( \left(1619, 65340\right) \), \( \left(1619, -66960\right) \), \( \left(3119, 173040\right) \), \( \left(3119, -176160\right) \), \( \left(5399, 394200\right) \), \( \left(5399, -399600\right) \), \( \left(7989, 710180\right) \), \( \left(7989, -718170\right) \), \( \left(21023, 3037680\right) \), \( \left(21023, -3058704\right) \), \( \left(22199, 3296400\right) \), \( \left(22199, -3318600\right) \), \( \left(29165399, 157493149200\right) \), \( \left(29165399, -157522314600\right) \)  Toggle raw display

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 374850 \)  =  \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7^{2} \cdot 17\)
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: \(-5878752997824000000 \)  =  \(-1 \cdot 2^{12} \cdot 3^{8} \cdot 5^{6} \cdot 7^{7} \cdot 17 \)
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( \frac{103823}{4386816} \)  =  \(2^{-12} \cdot 3^{-2} \cdot 7^{-1} \cdot 17^{-1} \cdot 47^{3}\)
Endomorphism ring: \(\Z\)
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$
Faltings height: \(2.2803828649914784396605186961\dots\)
Stable Faltings height: \(-0.046597310087283245890159960698\dots\)

BSD invariants

sage: E.rank()
 
magma: Rank(E);
 
Analytic rank: \(2\)
sage: E.regulator()
 
magma: Regulator(E);
 
Regulator: \(0.64194783703595641577985945986\dots\)
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: \(0.18941950283684838260936940179\dots\)
sage: E.tamagawa_numbers()
 
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
 
magma: TamagawaNumbers(E);
 
Tamagawa product: \( 768 \)  = \( ( 2^{2} \cdot 3 )\cdot2^{2}\cdot2^{2}\cdot2^{2}\cdot1 \)
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
magma: Order(TorsionSubgroup(E));
 
Torsion order: \(2\)
sage: E.sha().an_numerical()
 
magma: MordellWeilShaInformation(E);
 
Analytic order of Ш: \(1\) (rounded)

Modular invariants

Modular form 374850.2.a.la

sage: E.q_eigenform(20)
 
gp: xy = elltaniyama(E);
 
gp: x*deriv(xy[1])/(2*xy[2]+E.a1*xy[1]+E.a3)
 
magma: ModularForm(E);
 

\( q + q^{2} + q^{4} + q^{8} - 6q^{13} + q^{16} - q^{17} + O(q^{20}) \)  Toggle raw display

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 9437184
\( \Gamma_0(N) \)-optimal: yes
Manin constant: 1

Special L-value

sage: r = E.rank();
 
sage: E.lseries().dokchitser().derivative(1,r)/r.factorial()
 
gp: ar = ellanalyticrank(E);
 
gp: ar[2]/factorial(ar[1])
 
magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
 

\( L^{(2)}(E,1)/2! \) ≈ \( 23.346708506599877666659025725252542259 \)

Local data

This elliptic curve is not semistable. There are 5 primes of bad reduction:

sage: E.local_data()
 
gp: ellglobalred(E)[5]
 
magma: [LocalInformation(E,p) : p in BadPrimes(E)];
 
prime Tamagawa number Kodaira symbol Reduction type Root number ord(\(N\)) ord(\(\Delta\)) ord\((j)_{-}\)
\(2\) \(12\) \(I_{12}\) Split multiplicative -1 1 12 12
\(3\) \(4\) \(I_2^{*}\) Additive -1 2 8 2
\(5\) \(4\) \(I_0^{*}\) Additive 1 2 6 0
\(7\) \(4\) \(I_1^{*}\) Additive -1 2 7 1
\(17\) \(1\) \(I_{1}\) 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 X34.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^3\Z_2)$ generated by $\left(\begin{array}{rr} 5 & 0 \\ 4 & 3 \end{array}\right),\left(\begin{array}{rr} 7 & 7 \\ 4 & 3 \end{array}\right),\left(\begin{array}{rr} 3 & 0 \\ 0 & 3 \end{array}\right),\left(\begin{array}{rr} 7 & 0 \\ 4 & 3 \end{array}\right)$ and has index 12.

sage: rho = E.galois_representation();
 
sage: [rho.image_type(p) for p in rho.non_surjective()]
 
magma: [GaloisRepresentation(E,p): p in PrimesUpTo(20)];
 

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

sage: [E.padic_regulator(p) for p in primes(5,20) if E.conductor().valuation(p)<2]
 

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

No Iwasawa invariant data is available for this curve.

Isogenies

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2 and 4.
Its isogeny class 374850la consists of 4 curves linked by isogenies of degrees dividing 4.

Growth of torsion in number fields

The number fields $K$ of degree less than 24 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
$2$ \(\Q(\sqrt{-119}) \) \(\Z/2\Z \times \Z/2\Z\) Not in database
$2$ \(\Q(\sqrt{105}) \) \(\Z/4\Z\) Not in database
$2$ \(\Q(\sqrt{-255}) \) \(\Z/4\Z\) Not in database
$4$ \(\Q(\sqrt{105}, \sqrt{-119})\) \(\Z/2\Z \times \Z/4\Z\) Not in database
$8$ Deg 8 \(\Z/2\Z \times \Z/4\Z\) Not in database
$8$ 8.4.143884823040000.4 \(\Z/8\Z\) Not in database
$8$ Deg 8 \(\Z/8\Z\) Not in database
$8$ Deg 8 \(\Z/6\Z\) Not in database
$16$ Deg 16 \(\Z/4\Z \times \Z/4\Z\) Not in database
$16$ Deg 16 \(\Z/2\Z \times \Z/8\Z\) Not in database
$16$ Deg 16 \(\Z/2\Z \times \Z/8\Z\) Not in database
$16$ Deg 16 \(\Z/2\Z \times \Z/6\Z\) Not in database
$16$ Deg 16 \(\Z/12\Z\) Not in database
$16$ Deg 16 \(\Z/12\Z\) Not in database

We only show fields where the torsion growth is primitive. For fields not in the database, click on the degree shown to reveal the defining polynomial.