Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy=x^3+4224x+17842176\) | (homogenize, simplify) |
\(y^2z+xyz=x^3+4224xz^2+17842176z^3\) | (dehomogenize, simplify) |
\(y^2=x^3+5474277x+832428140598\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{6}\Z\)
Torsion generators
\( \left(528, 12672\right) \)
Integral points
\( \left(-256, 128\right) \), \( \left(0, 4224\right) \), \( \left(0, -4224\right) \), \( \left(528, 12672\right) \), \( \left(528, -13200\right) \)
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: | $-137523678664458240 $ | = | $-1 \cdot 2^{24} \cdot 3^{3} \cdot 5 \cdot 7^{4} \cdot 11^{3} \cdot 19 $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{8334681620170751}{137523678664458240} \) | = | $2^{-24} \cdot 3^{-3} \cdot 5^{-1} \cdot 7^{-4} \cdot 11^{-3} \cdot 19^{-1} \cdot 202751^{3}$ | 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);
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Sato-Tate group: | $\mathrm{SU}(2)$ | |||
Faltings height: | $1.9674500201168815597581662968\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $1.9674500201168815597581662968\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $1.0118506382749122\dots$ | |||
Szpiro ratio: | $4.389121371878548\dots$ |
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: | $0.25857628357922603958578293419\dots$ | 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: | $ 864 $ = $ ( 2^{3} \cdot 3 )\cdot3\cdot1\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)
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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) $ ≈ $ 6.2058308059014249500587904205 $ | 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
$\displaystyle 6.205830806 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.258576 \cdot 1.000000 \cdot 864}{6^2} \approx 6.205830806$
Modular invariants
Modular form 43890.2.a.ct
For more coefficients, see the Downloads section to the right.
Modular degree: | 497664 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | yes | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is semistable. There are 6 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $24$ | $I_{24}$ | Split multiplicative | -1 | 1 | 24 | 24 |
$3$ | $3$ | $I_{3}$ | Split multiplicative | -1 | 1 | 3 | 3 |
$5$ | $1$ | $I_{1}$ | Non-split multiplicative | 1 | 1 | 1 | 1 |
$7$ | $4$ | $I_{4}$ | Split multiplicative | -1 | 1 | 4 | 4 |
$11$ | $3$ | $I_{3}$ | Split multiplicative | -1 | 1 | 3 | 3 |
$19$ | $1$ | $I_{1}$ | 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 \( 175560 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 19 \), index $384$, genus $5$, and generators
$\left(\begin{array}{rr} 95776 & 3 \\ 127221 & 175474 \end{array}\right),\left(\begin{array}{rr} 1 & 24 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 24 & 1 \end{array}\right),\left(\begin{array}{rr} 125401 & 24 \\ 100332 & 289 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 12 & 145 \end{array}\right),\left(\begin{array}{rr} 83176 & 21 \\ 175275 & 175186 \end{array}\right),\left(\begin{array}{rr} 175537 & 24 \\ 175536 & 25 \end{array}\right),\left(\begin{array}{rr} 105352 & 3 \\ 104877 & 175474 \end{array}\right),\left(\begin{array}{rr} 43897 & 24 \\ 43794 & 175231 \end{array}\right),\left(\begin{array}{rr} 80481 & 51208 \\ 36116 & 7229 \end{array}\right),\left(\begin{array}{rr} 16 & 87801 \\ 146015 & 116666 \end{array}\right),\left(\begin{array}{rr} 15 & 106 \\ 174254 & 166331 \end{array}\right)$.
The torsion field $K:=\Q(E[175560])$ is a degree-$301950346199040000$ 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, 3, 4, 6 and 12.
Its isogeny class 43890.ct
consists of 8 curves linked by isogenies of
degrees dividing 12.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{6}\Z$ are as follows:
$[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
---|---|---|---|
$2$ | \(\Q(\sqrt{-3135}) \) | \(\Z/2\Z \oplus \Z/6\Z\) | Not in database |
$2$ | \(\Q(\sqrt{57}) \) | \(\Z/12\Z\) | Not in database |
$2$ | \(\Q(\sqrt{-55}) \) | \(\Z/12\Z\) | Not in database |
$4$ | \(\Q(\sqrt{-55}, \sqrt{57})\) | \(\Z/2\Z \oplus \Z/12\Z\) | Not in database |
$6$ | 6.0.5280199666875.2 | \(\Z/3\Z \oplus \Z/6\Z\) | 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 |
$9$ | 9.3.318157002959141467570680000.1 | \(\Z/18\Z\) | Not in database |
$12$ | deg 12 | \(\Z/6\Z \oplus \Z/6\Z\) | Not in database |
$12$ | deg 12 | \(\Z/3\Z \oplus \Z/12\Z\) | Not in database |
$12$ | deg 12 | \(\Z/3\Z \oplus \Z/12\Z\) | 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 |
$18$ | 18.0.28147693928214379809645817911841702951052939621806862535000000000000.1 | \(\Z/2\Z \oplus \Z/18\Z\) | Not in database |
$18$ | 18.6.6767289299044779518472322863853752775610023049570395200000000.1 | \(\Z/36\Z\) | Not in database |
$18$ | 18.0.421028069768801050147172478307413199837996958295000000000000.1 | \(\Z/36\Z\) | 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 | 19 |
---|---|---|---|---|---|---|
Reduction type | split | split | nonsplit | split | split | split |
$\lambda$-invariant(s) | 4 | 3 | 0 | 1 | 1 | 1 |
$\mu$-invariant(s) | 0 | 0 | 0 | 0 | 0 | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ of good reduction are zero.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.