Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy=x^3-x^2+37986291x-7986349035\) | (homogenize, simplify) |
\(y^2z+xyz=x^3-x^2z+37986291xz^2-7986349035z^3\) | (dehomogenize, simplify) |
\(y^2=x^3+607780653x-510518557586\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{6}\Z\)
Torsion generators
\( \left(16086, 2174907\right) \)
Integral points
\( \left(210, -105\right) \), \( \left(2611, 328832\right) \), \( \left(2611, -331443\right) \), \( \left(16086, 2174907\right) \), \( \left(16086, -2190993\right) \)
Invariants
Conductor: | \( 90090 \) | = | $2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11 \cdot 13$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $-3535496761931195811750000 $ | = | $-1 \cdot 2^{4} \cdot 3^{10} \cdot 5^{6} \cdot 7^{12} \cdot 11^{3} \cdot 13 $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{8315279469612171276463151}{4849789796887785750000} \) | = | $2^{-4} \cdot 3^{-4} \cdot 5^{-6} \cdot 7^{-12} \cdot 11^{-3} \cdot 13^{-1} \cdot 59^{3} \cdot 3433789^{3}$ | 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: | $3.3998346405009235083881850260\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $2.8505284961668686626905624075\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $1.0302632900467197\dots$ | |||
Szpiro ratio: | $5.607349721220169\dots$ |
BSD invariants
Analytic rank: | $0$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $1$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.046657944312074529168314496487\dots$ | 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);
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Tamagawa product: | $ 1728 $ = $ 2\cdot2^{2}\cdot( 2 \cdot 3 )\cdot( 2^{2} \cdot 3 )\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)
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Torsion order: | $6$ | 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: | $1$ ( exact) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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Special value: | $ L(E,1) $ ≈ $ 2.2395813269795774000790958314 $ | 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
$\displaystyle 2.239581327 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.046658 \cdot 1.000000 \cdot 1728}{6^2} \approx 2.239581327$
Modular invariants
Modular form 90090.2.a.ce
For more coefficients, see the Downloads section to the right.
Modular degree: | 18579456 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | no | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 6 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $2$ | $I_{4}$ | Non-split multiplicative | 1 | 1 | 4 | 4 |
$3$ | $4$ | $I_{4}^{*}$ | Additive | -1 | 2 | 10 | 4 |
$5$ | $6$ | $I_{6}$ | Split multiplicative | -1 | 1 | 6 | 6 |
$7$ | $12$ | $I_{12}$ | Split multiplicative | -1 | 1 | 12 | 12 |
$11$ | $3$ | $I_{3}$ | Split multiplicative | -1 | 1 | 3 | 3 |
$13$ | $1$ | $I_{1}$ | Split multiplicative | -1 | 1 | 1 | 1 |
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 | 4.6.0.1 |
$3$ | 3B.1.1 | 3.8.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 24024 = 2^{3} \cdot 3 \cdot 7 \cdot 11 \cdot 13 \), index $384$, genus $5$, and generators
$\left(\begin{array}{rr} 21856 & 3 \\ 12645 & 23938 \end{array}\right),\left(\begin{array}{rr} 18025 & 24 \\ 18510 & 1687 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 12 & 145 \end{array}\right),\left(\begin{array}{rr} 3433 & 24 \\ 17172 & 289 \end{array}\right),\left(\begin{array}{rr} 24001 & 24 \\ 24000 & 25 \end{array}\right),\left(\begin{array}{rr} 1 & 24 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 14800 & 3 \\ 19869 & 23938 \end{array}\right),\left(\begin{array}{rr} 15 & 106 \\ 22718 & 14795 \end{array}\right),\left(\begin{array}{rr} 20019 & 16012 \\ 24016 & 11979 \end{array}\right),\left(\begin{array}{rr} 13017 & 17020 \\ 7052 & 11045 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 24 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[24024])$ is a degree-$133905855283200$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/24024\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3, 4, 6 and 12.
Its isogeny class 90090.ce
consists of 8 curves linked by isogenies of
degrees dividing 12.
Twists
The minimal quadratic twist of this elliptic curve is 30030.bt8, its twist by $-3$.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{6}\Z$ are as follows:
$[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
---|---|---|---|
$2$ | \(\Q(\sqrt{-143}) \) | \(\Z/2\Z \oplus \Z/6\Z\) | Not in database |
$2$ | \(\Q(\sqrt{429}) \) | \(\Z/12\Z\) | Not in database |
$2$ | \(\Q(\sqrt{-3}) \) | \(\Z/12\Z\) | Not in database |
$4$ | \(\Q(\sqrt{-3}, \sqrt{-143})\) | \(\Z/2\Z \oplus \Z/12\Z\) | Not in database |
$6$ | 6.0.80951927472.3 | \(\Z/3\Z \oplus \Z/12\Z\) | Not in database |
$8$ | deg 8 | \(\Z/2\Z \oplus \Z/12\Z\) | Not in database |
$8$ | deg 8 | \(\Z/24\Z\) | Not in database |
$8$ | deg 8 | \(\Z/24\Z\) | Not in database |
$9$ | 9.3.21353869767453342770400750000.2 | \(\Z/18\Z\) | Not in database |
$12$ | deg 12 | \(\Z/6\Z \oplus \Z/12\Z\) | Not in database |
$16$ | deg 16 | \(\Z/4\Z \oplus \Z/12\Z\) | Not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/24\Z\) | Not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/24\Z\) | Not in database |
$18$ | 18.0.225345036407637636931920990857976003384330635742217227937500000000.1 | \(\Z/2\Z \oplus \Z/18\Z\) | Not in database |
$18$ | 18.6.676035109222912910795762972573928010152991907226651683812500000000.1 | \(\Z/36\Z\) | Not in database |
$18$ | 18.0.1367963262136073637741308146397175381390946801687500000000.1 | \(\Z/36\Z\) | 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 |
---|---|---|---|---|---|---|
Reduction type | nonsplit | add | split | split | split | split |
$\lambda$-invariant(s) | 5 | - | 1 | 5 | 1 | 1 |
$\mu$-invariant(s) | 0 | - | 0 | 0 | 0 | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ of good reduction are zero.
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.