Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy=x^3-x^2-46061492x-112771265584\) | (homogenize, simplify) |
\(y^2z+xyz=x^3-x^2z-46061492xz^2-112771265584z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-736983875x-7218097981250\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = | \(\left(-3320, 61244\right)\) |
$\hat{h}(P)$ | ≈ | $1.5534071923595450122063369992$ |
Torsion generators
\( \left(-\frac{12349}{4}, \frac{12349}{8}\right) \)
Integral points
\( \left(-3320, 61244\right) \), \( \left(-3320, -57924\right) \), \( \left(71944, 19172028\right) \), \( \left(71944, -19243972\right) \)
Invariants
Conductor: | \( 126350 \) | = | $2 \cdot 5^{2} \cdot 7 \cdot 19^{2}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $759499111293272000000000 $ | = | $2^{12} \cdot 5^{9} \cdot 7^{12} \cdot 19^{3} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{806764685224507983}{56693912375296} \) | = | $2^{-12} \cdot 3^{3} \cdot 7^{-12} \cdot 227^{3} \cdot 1367^{3}$ | 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: | $3.3297320908915462875120862430\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $1.3865439117743608915592598851\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $1.0562630714144663\dots$ | |||
Szpiro ratio: | $5.495114032062656\dots$ |
BSD invariants
Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $1.5534071923595450122063369992\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.058208121551012233665628544870\dots$ | 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: | $ 96 $ = $ 2\cdot2\cdot( 2^{2} \cdot 3 )\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'(E,1) $ ≈ $ 2.1701019521059449223137251063 $ | 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
$\displaystyle 2.170101952 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.058208 \cdot 1.553407 \cdot 96}{2^2} \approx 2.170101952$
Modular invariants
Modular form 126350.2.a.ba
For more coefficients, see the Downloads section to the right.
Modular degree: | 22809600 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | no | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 4 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $2$ | $I_{12}$ | Non-split multiplicative | 1 | 1 | 12 | 12 |
$5$ | $2$ | $III^{*}$ | Additive | -1 | 2 | 9 | 0 |
$7$ | $12$ | $I_{12}$ | Split multiplicative | -1 | 1 | 12 | 12 |
$19$ | $2$ | $III$ | 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 | 8.12.0.29 |
$3$ | 3Nn | 3.3.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 31920 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \cdot 19 \), index $576$, genus $37$, and generators
$\left(\begin{array}{rr} 6752 & 25 \\ 13111 & 2368 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 12 & 145 \end{array}\right),\left(\begin{array}{rr} 10657 & 16 \\ 21536 & 241 \end{array}\right),\left(\begin{array}{rr} 21296 & 23961 \\ 3417 & 20528 \end{array}\right),\left(\begin{array}{rr} 7 & 24 \\ 31824 & 31591 \end{array}\right),\left(\begin{array}{rr} 9 & 16 \\ 31736 & 31593 \end{array}\right),\left(\begin{array}{rr} 6368 & 31899 \\ 13341 & 752 \end{array}\right),\left(\begin{array}{rr} 1 & 24 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 31897 & 24 \\ 31896 & 25 \end{array}\right),\left(\begin{array}{rr} 6687 & 6698 \\ 28450 & 423 \end{array}\right),\left(\begin{array}{rr} 13681 & 48 \\ 4572 & 577 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 24 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[31920])$ is a degree-$244000279756800$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/31920\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 126350bm
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 126350db2, its twist by $5$.
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 |
---|---|---|---|
$2$ | \(\Q(\sqrt{95}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | Not in database |
$4$ | 4.0.3429500.2 | \(\Z/4\Z\) | Not in database |
$8$ | 8.4.48174982144000000.4 | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$8$ | 8.0.188183524000000.14 | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$8$ | 8.2.1607645964796875.1 | \(\Z/6\Z\) | Not in database |
$16$ | deg 16 | \(\Z/4\Z \oplus \Z/4\Z\) | Not in database |
$16$ | deg 16 | \(\Z/8\Z\) | Not in database |
$16$ | deg 16 | \(\Z/3\Z \oplus \Z/6\Z\) | 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 | ss | add | split | ord | ord | ord | add | ord | ss | ss | ord | ord | ss | ord |
$\lambda$-invariant(s) | 4 | 1,1 | - | 2 | 1 | 1 | 1 | - | 1 | 1,1 | 1,1 | 1 | 1 | 1,1 | 1 |
$\mu$-invariant(s) | 0 | 0,0 | - | 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
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.