Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy+y=x^3-x^2+7120x+57747\) | (homogenize, simplify) |
\(y^2z+xyz+yz^2=x^3-x^2z+7120xz^2+57747z^3\) | (dehomogenize, simplify) |
\(y^2=x^3+113925x+3809750\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z\)
Infinite order Mordell-Weil generators and heights
$P$ | = | \(\left(149, 2025\right)\) | \(\left(-\frac{59}{9}, \frac{2875}{27}\right)\) |
$\hat{h}(P)$ | ≈ | $0.14594252241243860527243331569$ | $1.9578396296002047963656165888$ |
Integral points
\( \left(-1, 225\right) \), \( \left(-1, -225\right) \), \( \left(9, 345\right) \), \( \left(9, -355\right) \), \( \left(23, 471\right) \), \( \left(23, -495\right) \), \( \left(59, 795\right) \), \( \left(59, -855\right) \), \( \left(93, 1185\right) \), \( \left(93, -1279\right) \), \( \left(149, 2025\right) \), \( \left(149, -2175\right) \), \( \left(249, 4025\right) \), \( \left(249, -4275\right) \), \( \left(485, 10593\right) \), \( \left(485, -11079\right) \), \( \left(849, 24425\right) \), \( \left(849, -25275\right) \), \( \left(3299, 187875\right) \), \( \left(3299, -191175\right) \), \( \left(3509, 206145\right) \), \( \left(3509, -209655\right) \), \( \left(3574749, 6756992225\right) \), \( \left(3574749, -6760566975\right) \)
Invariants
Conductor: | \( 418950 \) | = | $2 \cdot 3^{2} \cdot 5^{2} \cdot 7^{2} \cdot 19$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
|
Discriminant: | $-24634260000000 $ | = | $-1 \cdot 2^{8} \cdot 3^{3} \cdot 5^{7} \cdot 7^{4} \cdot 19 $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
|
j-invariant: | \( \frac{39413493}{24320} \) | = | $2^{-8} \cdot 3^{3} \cdot 5^{-1} \cdot 7^{2} \cdot 19^{-1} \cdot 31^{3}$ | comment: j-invariant
sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
oscar: j_invariant(E)
|
Endomorphism ring: | $\Z$ | |||
Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | sage: E.has_cm()
magma: HasComplexMultiplication(E);
| |
Sato-Tate group: | $\mathrm{SU}(2)$ | |||
Faltings height: | $1.2587350375692762338998291684\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
|
||
Stable Faltings height: | $-0.46927370716657247795114605526\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
|
||
$abc$ quality: | $0.8711545350577963\dots$ | |||
Szpiro ratio: | $2.9528178485147576\dots$ |
BSD invariants
Analytic rank: | $2$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
|
Regulator: | $0.28512823476593729303561412834\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
|
Real period: | $0.41527363280991730646767275997\dots$ | comment: Real Period
sage: E.period_lattice().omega()
gp: if(E.disc>0,2,1)*E.omega[1]
magma: (Discriminant(E) gt 0 select 2 else 1) * RealPeriod(E);
|
Tamagawa product: | $ 192 $ = $ 2^{3}\cdot2\cdot2^{2}\cdot3\cdot1 $ | comment: Tamagawa numbers
sage: E.tamagawa_numbers()
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
magma: TamagawaNumbers(E);
oscar: tamagawa_numbers(E)
|
Torsion order: | $1$ | comment: Torsion order
sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
oscar: prod(torsion_structure(E)[1])
|
Analytic order of Ш: | $1$ ( rounded) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
|
Special value: | $ L^{(2)}(E,1)/2! $ ≈ $ 22.733997670642510360830329447 $ | comment: Special L-value
r = E.rank();
gp: [r,L1r] = ellanalyticrank(E); L1r/r!
magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
|
BSD formula
$\displaystyle 22.733997671 \approx L^{(2)}(E,1)/2! \overset{?}{=} \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.415274 \cdot 0.285128 \cdot 192}{1^2} \approx 22.733997671$
Modular invariants
Modular form 418950.2.a.ju
For more coefficients, see the Downloads section to the right.
Modular degree: | 1032192 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
|
$ \Gamma_0(N) $-optimal: | yes | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
|
Local data
This elliptic curve is not semistable. There are 5 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $8$ | $I_{8}$ | Split multiplicative | -1 | 1 | 8 | 8 |
$3$ | $2$ | $III$ | Additive | 1 | 2 | 3 | 0 |
$5$ | $4$ | $I_{1}^{*}$ | Additive | 1 | 2 | 7 | 1 |
$7$ | $3$ | $IV$ | Additive | 1 | 2 | 4 | 0 |
$19$ | $1$ | $I_{1}$ | Non-split multiplicative | 1 | 1 | 1 | 1 |
Galois representations
The $\ell$-adic Galois representation has maximal image for all primes $\ell$.
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1140 = 2^{2} \cdot 3 \cdot 5 \cdot 19 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 781 & 2 \\ 781 & 3 \end{array}\right),\left(\begin{array}{rr} 457 & 2 \\ 457 & 3 \end{array}\right),\left(\begin{array}{rr} 1139 & 2 \\ 1138 & 3 \end{array}\right),\left(\begin{array}{rr} 571 & 2 \\ 571 & 3 \end{array}\right),\left(\begin{array}{rr} 761 & 2 \\ 761 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 1139 & 0 \end{array}\right)$.
The torsion field $K:=\Q(E[1140])$ is a degree-$136160870400$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1140\Z)$.
Isogenies
This curve has no rational isogenies. Its isogeny class 418950ju consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 83790a1, its twist by $5$.
Iwasawa invariants
No Iwasawa invariant data is available for this curve.
$p$-adic regulators
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.