Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy=x^3+x^2-478x+3232\) | (homogenize, simplify) |
\(y^2z+xyz=x^3+x^2z-478xz^2+3232z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-620163x+160091262\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generators and heights
$P$ | = | \(\left(1, 52\right)\) | \(\left(22, 52\right)\) |
$\hat{h}(P)$ | ≈ | $0.74040856292430427159211786413$ | $1.3471121354557005283082105783$ |
Torsion generators
\( \left(8, -4\right) \), \( \left(16, -8\right) \)
Integral points
\( \left(-24, 52\right) \), \( \left(-24, -28\right) \), \( \left(-17, 91\right) \), \( \left(-17, -74\right) \), \( \left(-11, 91\right) \), \( \left(-11, -80\right) \), \( \left(-6, 80\right) \), \( \left(-6, -74\right) \), \( \left(1, 52\right) \), \( \left(1, -53\right) \), \( \left(6, 22\right) \), \( \left(6, -28\right) \), \( \left(8, -4\right) \), \( \left(16, -8\right) \), \( \left(17, 11\right) \), \( \left(17, -28\right) \), \( \left(22, 52\right) \), \( \left(22, -74\right) \), \( \left(27, 91\right) \), \( \left(27, -118\right) \), \( \left(46, 262\right) \), \( \left(46, -308\right) \), \( \left(71, 542\right) \), \( \left(71, -613\right) \), \( \left(141, 1592\right) \), \( \left(141, -1733\right) \), \( \left(232, 3412\right) \), \( \left(232, -3644\right) \), \( \left(346, 6262\right) \), \( \left(346, -6608\right) \), \( \left(1072, 34576\right) \), \( \left(1072, -35648\right) \)
Invariants
Conductor: | \( 43890 \) | = | $2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 19$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $1926332100 $ | = | $2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2} \cdot 11^{2} \cdot 19^{2} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{12117869279209}{1926332100} \) | = | $2^{-2} \cdot 3^{-2} \cdot 5^{-2} \cdot 7^{-2} \cdot 11^{-2} \cdot 19^{-2} \cdot 103^{3} \cdot 223^{3}$ | 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: | $0.50414610731126930054960932568\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $0.50414610731126930054960932568\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $0.8454293478375657\dots$ | |||
Szpiro ratio: | $2.8182671173601626\dots$ |
BSD invariants
Analytic rank: | $2$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $0.96596748158078547363212456643\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $1.4145021168571855769714426680\dots$ | 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: | $ 64 $ = $ 2\cdot2\cdot2\cdot2\cdot2\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: | $4$ | 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! $ ≈ $ 5.4654521900449018817167176763 $ | 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
$\displaystyle 5.465452190 \approx L^{(2)}(E,1)/2! \overset{?}{=} \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 1.414502 \cdot 0.965967 \cdot 64}{4^2} \approx 5.465452190$
Modular invariants
Modular form 43890.2.a.a
For more coefficients, see the Downloads section to the right.
Modular degree: | 26624 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | no | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is semistable. There are 6 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $2$ | $I_{2}$ | Non-split multiplicative | 1 | 1 | 2 | 2 |
$3$ | $2$ | $I_{2}$ | Non-split multiplicative | 1 | 1 | 2 | 2 |
$5$ | $2$ | $I_{2}$ | Non-split multiplicative | 1 | 1 | 2 | 2 |
$7$ | $2$ | $I_{2}$ | Non-split multiplicative | 1 | 1 | 2 | 2 |
$11$ | $2$ | $I_{2}$ | Non-split multiplicative | 1 | 1 | 2 | 2 |
$19$ | $2$ | $I_{2}$ | Split multiplicative | -1 | 1 | 2 | 2 |
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 |
---|---|---|
$2$ | 2Cs | 8.12.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 175560 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 19 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 140451 & 2 \\ 140446 & 175559 \end{array}\right),\left(\begin{array}{rr} 127683 & 2 \\ 31918 & 175559 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 125401 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 147841 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 87781 & 4 \\ 2 & 9 \end{array}\right),\left(\begin{array}{rr} 175557 & 4 \\ 175556 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 43893 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 3 & 2 \\ 117038 & 175559 \end{array}\right)$.
The torsion field $K:=\Q(E[175560])$ is a degree-$2415602769592320000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/175560\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 43890a
consists of 4 curves linked by isogenies of
degrees dividing 4.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \oplus \Z/{2}\Z$ are as follows:
$[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
---|---|---|---|
$4$ | \(\Q(\sqrt{-2}, \sqrt{133})\) | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$4$ | \(\Q(\sqrt{-133}, \sqrt{-165})\) | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$4$ | \(\Q(\sqrt{2}, \sqrt{165})\) | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$8$ | deg 8 | \(\Z/2\Z \oplus \Z/6\Z\) | 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 | \(\Z/2\Z \oplus \Z/8\Z\) | 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 | nonsplit | nonsplit | nonsplit | ord | ord | split | ss | ord | ord | ord | ord | ord | ss |
$\lambda$-invariant(s) | 2 | 2 | 4 | 2 | 2 | 2 | 4 | 3 | 2,2 | 2 | 2 | 2 | 2 | 2 | 2,2 |
$\mu$-invariant(s) | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0,0 | 0 | 0 | 0 | 0 | 0 | 0,0 |
$p$-adic regulators
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.