Properties

Label 377520.dy2
Conductor $377520$
Discriminant $2.967\times 10^{26}$
j-invariant \( \frac{287849398425814280018}{81784533026485575} \)
CM no
Rank $1$
Torsion structure \(\Z/{2}\Z\)

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Show commands: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

Minimal Weierstrass equation

Simplified equation

\(y^2=x^3+x^2-211369616x+843820298484\) Copy content Toggle raw display (homogenize, simplify)
\(y^2z=x^3+x^2z-211369616xz^2+843820298484z^3\) Copy content Toggle raw display (dehomogenize, simplify)
\(y^2=x^3-17120938923x+615196360411578\) Copy content Toggle raw display (homogenize, minimize)

sage: E = EllipticCurve([0, 1, 0, -211369616, 843820298484])
 
gp: E = ellinit([0, 1, 0, -211369616, 843820298484])
 
magma: E := EllipticCurve([0, 1, 0, -211369616, 843820298484]);
 
sage: E.short_weierstrass_model()
 
magma: WeierstrassModel(E);
 

Mordell-Weil group structure

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

Infinite order Mordell-Weil generator and height

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

$P$ =  \(\left(3454, 393660\right)\) Copy content Toggle raw display
$\hat{h}(P)$ ≈  $0.74799834049456666612945269031$

Torsion generators

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

\( \left(-16229, 0\right) \) Copy content Toggle raw display

Integral points

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

\( \left(-16229, 0\right) \), \((-6428,\pm 1391742)\), \((-5066,\pm 1335900)\), \((3454,\pm 393660)\), \((13174,\pm 588060)\), \((42820,\pm 8384958)\) Copy content Toggle raw display

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 377520 \)  =  $2^{4} \cdot 3 \cdot 5 \cdot 11^{2} \cdot 13$
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: $296727120103288446428313600 $  =  $2^{11} \cdot 3^{28} \cdot 5^{2} \cdot 11^{7} \cdot 13 $
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( \frac{287849398425814280018}{81784533026485575} \)  =  $2 \cdot 3^{-28} \cdot 5^{-2} \cdot 11^{-1} \cdot 13^{-1} \cdot 37^{3} \cdot 141637^{3}$
Endomorphism ring: $\Z$
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$
Faltings height: $3.7862937703109815045008313510\dots$
Stable Faltings height: $1.9519612183985130321707301173\dots$

BSD invariants

sage: E.rank()
 
magma: Rank(E);
 
Analytic rank: $1$
sage: E.regulator()
 
magma: Regulator(E);
 
Regulator: $0.74799834049456666612945269031\dots$
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: $0.050864126098739793226663538513\dots$
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: $ 896 $  = $ 2^{2}\cdot( 2^{2} \cdot 7 )\cdot2\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: $1$ (exact)
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);
 
Special value: $ L'(E,1) $ ≈ $ 8.5223671484142783638433844459 $

Modular invariants

Modular form 377520.2.a.dy

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);
 

\( q + q^{3} - q^{5} - 4 q^{7} + q^{9} - q^{13} - q^{15} + 2 q^{17} + 4 q^{19} + O(q^{20}) \) Copy content Toggle raw display

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 165150720
$ \Gamma_0(N) $-optimal: no
Manin constant: 1

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$(j)_{-}$
$2$ $4$ $I_{3}^{*}$ Additive 1 4 11 0
$3$ $28$ $I_{28}$ Split multiplicative -1 1 28 28
$5$ $2$ $I_{2}$ Non-split multiplicative 1 1 2 2
$11$ $4$ $I_{1}^{*}$ Additive -1 2 7 1
$13$ $1$ $I_{1}$ Non-split multiplicative 1 1 1 1

Galois representations

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 $\ell$-adic Galois representation has maximal image for all primes $\ell$ except those listed in the table below.

prime $\ell$ mod-$\ell$ image $\ell$-adic image
$2$ 2B 8.12.0.7

$p$-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 377520.dy 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
$2$ \(\Q(\sqrt{286}) \) \(\Z/2\Z \oplus \Z/2\Z\) Not in database
$2$ \(\Q(\sqrt{-143}) \) \(\Z/4\Z\) Not in database
$2$ \(\Q(\sqrt{-2}) \) \(\Z/4\Z\) Not in database
$4$ \(\Q(\sqrt{-2}, \sqrt{-143})\) \(\Z/2\Z \oplus \Z/4\Z\) Not in database
$8$ Deg 8 \(\Z/2\Z \oplus \Z/4\Z\) Not in database
$8$ 8.0.366349705989696.15 \(\Z/8\Z\) Not in database
$8$ Deg 8 \(\Z/8\Z\) Not in database
$8$ Deg 8 \(\Z/6\Z\) Not in database
$16$ Deg 16 \(\Z/4\Z \oplus \Z/4\Z\) Not in database
$16$ Deg 16 \(\Z/2\Z \oplus \Z/8\Z\) Not in database
$16$ Deg 16 \(\Z/2\Z \oplus \Z/8\Z\) Not in database
$16$ Deg 16 \(\Z/2\Z \oplus \Z/6\Z\) 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.