Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy=x^3+x^2-17798x-835392\) | (homogenize, simplify) |
\(y^2z+xyz=x^3+x^2z-17798xz^2-835392z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-23066883x-38630049282\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generators and heights
$P$ | = | \(\left(-91, 248\right)\) | \(\left(-64, 248\right)\) |
$\hat{h}(P)$ | ≈ | $2.1222684284368411599746869649$ | $2.5214385180815913609923748853$ |
Torsion generators
\( \left(-96, 48\right) \), \( \left(152, -76\right) \)
Integral points
\( \left(-96, 48\right) \), \( \left(-91, 248\right) \), \( \left(-91, -157\right) \), \( \left(-64, 248\right) \), \( \left(-64, -184\right) \), \( \left(152, -76\right) \), \( \left(154, 248\right) \), \( \left(154, -402\right) \), \( \left(183, 1350\right) \), \( \left(183, -1533\right) \), \( \left(394, 7118\right) \), \( \left(394, -7512\right) \), \( \left(1144, 37868\right) \), \( \left(1144, -39012\right) \), \( \left(1826, 76928\right) \), \( \left(1826, -78754\right) \), \( \left(26099, 4203338\right) \), \( \left(26099, -4229437\right) \)
Invariants
Conductor: | \( 28830 \) | = | $2 \cdot 3 \cdot 5 \cdot 31^{2}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $64699018344900 $ | = | $2^{2} \cdot 3^{6} \cdot 5^{2} \cdot 31^{6} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{702595369}{72900} \) | = | $2^{-2} \cdot 3^{-6} \cdot 5^{-2} \cdot 7^{3} \cdot 127^{3}$ | 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: | $1.3852798528448973028475858046\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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||
Stable Faltings height: | $-0.33171374939767582011699635767\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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BSD invariants
Analytic rank: | $2$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $5.3191706846335516470011530329\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
|
Real period: | $0.41610673003599834435910371034\dots$ | 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: | $ 32 $ = $ 2\cdot2\cdot2\cdot2^{2} $ | 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! $ ≈ $ 4.4266854401724195242601042594 $ | 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
$\displaystyle 4.426685440 \approx L^{(2)}(E,1)/2! \overset{?}{=} \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.416107 \cdot 5.319171 \cdot 32}{4^2} \approx 4.426685440$
Modular invariants
Modular form 28830.2.a.a
For more coefficients, see the Downloads section to the right.
Modular degree: | 115200 | 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)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $2$ | $I_{2}$ | Non-split multiplicative | 1 | 1 | 2 | 2 |
$3$ | $2$ | $I_{6}$ | Non-split multiplicative | 1 | 1 | 6 | 6 |
$5$ | $2$ | $I_{2}$ | Non-split multiplicative | 1 | 1 | 2 | 2 |
$31$ | $4$ | $I_0^{*}$ | Additive | -1 | 2 | 6 | 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 |
---|---|---|
$2$ | 2Cs | 2.6.0.1 |
$3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 3720 = 2^{3} \cdot 3 \cdot 5 \cdot 31 \), index $384$, genus $5$, and generators
$\left(\begin{array}{rr} 2977 & 2046 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 2791 & 372 \\ 186 & 2233 \end{array}\right),\left(\begin{array}{rr} 187 & 2046 \\ 1674 & 1675 \end{array}\right),\left(\begin{array}{rr} 1919 & 0 \\ 0 & 3719 \end{array}\right),\left(\begin{array}{rr} 9 & 4 \\ 3704 & 3713 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 3709 & 12 \\ 3708 & 13 \end{array}\right),\left(\begin{array}{rr} 2481 & 124 \\ 2542 & 249 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[3720])$ is a degree-$82280448000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/3720\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 28830h
consists of 8 curves linked by isogenies of
degrees dividing 12.
Twists
The minimal quadratic twist of this elliptic curve is 30a2, its twist by $-31$.
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 |
---|---|---|---|
$2$ | \(\Q(\sqrt{-31}) \) | \(\Z/2\Z \oplus \Z/6\Z\) | Not in database |
$4$ | \(\Q(\sqrt{10}, \sqrt{-62})\) | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$4$ | \(\Q(\sqrt{6}, \sqrt{62})\) | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$4$ | \(\Q(\sqrt{-15}, \sqrt{-93})\) | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$6$ | 6.2.8043570000.4 | \(\Z/2\Z \oplus \Z/6\Z\) | Not in database |
$8$ | 8.0.2364213760000.27 | \(\Z/2\Z \oplus \Z/12\Z\) | Not in database |
$8$ | 8.0.11968832160000.3 | \(\Z/2\Z \oplus \Z/12\Z\) | Not in database |
$8$ | 8.0.306402103296.2 | \(\Z/2\Z \oplus \Z/12\Z\) | Not in database |
$12$ | deg 12 | \(\Z/6\Z \oplus \Z/6\Z\) | Not in database |
$12$ | deg 12 | \(\Z/2\Z \oplus \Z/12\Z\) | 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 |
$18$ | 18.0.544368373881532186745734947900000000.1 | \(\Z/2\Z \oplus \Z/18\Z\) | 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 | ord | ss | ord | ord | ord | ss | ord | add | ord | ord | ord | ss |
$\lambda$-invariant(s) | 16 | 2 | 2 | 6 | 2,2 | 2 | 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 | 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.