Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy=x^3+x^2-7192x+228844\) | (homogenize, simplify) |
\(y^2z+xyz=x^3+x^2z-7192xz^2+228844z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-9321507x+10816764894\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generators and heights
$P$ | = | \(\left(58, 76\right)\) | \(\left(43, 16\right)\) |
$\hat{h}(P)$ | ≈ | $0.14498104867699065106537908617$ | $1.1308114807532584030249841344$ |
Torsion generators
\( \left(44, -22\right) \)
Integral points
\( \left(-82, 566\right) \), \( \left(-82, -484\right) \), \( \left(-77, 616\right) \), \( \left(-77, -539\right) \), \( \left(-47, 706\right) \), \( \left(-47, -659\right) \), \( \left(-12, 566\right) \), \( \left(-12, -554\right) \), \( \left(-5, 517\right) \), \( \left(-5, -512\right) \), \( \left(18, 316\right) \), \( \left(18, -334\right) \), \( \left(30, 188\right) \), \( \left(30, -218\right) \), \( \left(43, 16\right) \), \( \left(43, -59\right) \), \( \left(44, -22\right) \), \( \left(53, -19\right) \), \( \left(53, -34\right) \), \( \left(58, 76\right) \), \( \left(58, -134\right) \), \( \left(70, 238\right) \), \( \left(70, -308\right) \), \( \left(93, 566\right) \), \( \left(93, -659\right) \), \( \left(135, 1252\right) \), \( \left(135, -1387\right) \), \( \left(148, 1486\right) \), \( \left(148, -1634\right) \), \( \left(268, 4066\right) \), \( \left(268, -4334\right) \), \( \left(333, 5741\right) \), \( \left(333, -6074\right) \), \( \left(394, 7468\right) \), \( \left(394, -7862\right) \), \( \left(863, 24821\right) \), \( \left(863, -25684\right) \), \( \left(1318, 47116\right) \), \( \left(1318, -48434\right) \), \( \left(23578, 3608716\right) \), \( \left(23578, -3632294\right) \), \( \left(48778, 10748716\right) \), \( \left(48778, -10797494\right) \)
Invariants
Conductor: | \( 35490 \) | = | $2 \cdot 3 \cdot 5 \cdot 7 \cdot 13^{2}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $593437162500 $ | = | $2^{2} \cdot 3^{2} \cdot 5^{5} \cdot 7^{4} \cdot 13^{3} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
|
j-invariant: | \( \frac{18729968230693}{270112500} \) | = | $2^{-2} \cdot 3^{-2} \cdot 5^{-5} \cdot 7^{-4} \cdot 26557^{3}$ | comment: j-invariant
sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
oscar: j_invariant(E)
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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: | $1.0630975846682017352373664347\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $0.42186024530281755122399457431\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $0.9525180123639401\dots$ | |||
Szpiro ratio: | $3.651424186609954\dots$ |
BSD invariants
Analytic rank: | $2$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
|
Regulator: | $0.16146861398824515049210333908\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.91961622007702926494909801875\dots$ | 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: | $ 160 $ = $ 2\cdot2\cdot5\cdot2^{2}\cdot2 $ | 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)
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Torsion order: | $2$ | comment: Torsion order
sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
oscar: prod(torsion_structure(E)[1])
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Analytic order of Ш: | $1$ ( rounded) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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Special value: | $ L^{(2)}(E,1)/2! $ ≈ $ 5.9395662582778775350697353103 $ | comment: Special L-value
r = E.rank();
gp: [r,L1r] = ellanalyticrank(E); L1r/r!
magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
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BSD formula
$\displaystyle 5.939566258 \approx L^{(2)}(E,1)/2! \overset{?}{=} \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.919616 \cdot 0.161469 \cdot 160}{2^2} \approx 5.939566258$
Modular invariants
Modular form 35490.2.a.p
For more coefficients, see the Downloads section to the right.
Modular degree: | 107520 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | yes | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 5 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $2$ | $I_{2}$ | Non-split multiplicative | 1 | 1 | 2 | 2 |
$3$ | $2$ | $I_{2}$ | Non-split multiplicative | 1 | 1 | 2 | 2 |
$5$ | $5$ | $I_{5}$ | Split multiplicative | -1 | 1 | 5 | 5 |
$7$ | $4$ | $I_{4}$ | Split multiplicative | -1 | 1 | 4 | 4 |
$13$ | $2$ | $III$ | Additive | -1 | 2 | 3 | 0 |
Galois representations
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 | 2.3.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 520 = 2^{3} \cdot 5 \cdot 13 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 329 & 196 \\ 64 & 455 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 314 & 1 \\ 103 & 0 \end{array}\right),\left(\begin{array}{rr} 517 & 4 \\ 516 & 5 \end{array}\right),\left(\begin{array}{rr} 164 & 1 \\ 479 & 0 \end{array}\right),\left(\begin{array}{rr} 261 & 4 \\ 2 & 9 \end{array}\right)$.
The torsion field $K:=\Q(E[520])$ is a degree-$1610219520$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/520\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 35490w
consists of 2 curves linked by isogenies of
degree 2.
Twists
This elliptic curve is its own minimal quadratic twist.
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 |
---|---|---|---|
$2$ | \(\Q(\sqrt{65}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | Not in database |
$4$ | 4.0.703040.1 | \(\Z/4\Z\) | Not in database |
$8$ | 8.4.6255544464000000.9 | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$8$ | 8.0.12356631040000.19 | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$8$ | deg 8 | \(\Z/6\Z\) | Not in database |
$16$ | deg 16 | \(\Z/8\Z\) | Not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | 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.
Iwasawa invariants
$p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Reduction type | nonsplit | nonsplit | split | split | ord | add | ord | ord | ord | ord | ord | ord | ord | ord | ord |
$\lambda$-invariant(s) | 4 | 6 | 3 | 9 | 2 | - | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
$\mu$-invariant(s) | 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.
$p$-adic regulators
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.