Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy+y=x^3-276x-3586\) | (homogenize, simplify) |
\(y^2z+xyz+yz^2=x^3-276xz^2-3586z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-357075x-166225554\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generators and heights
$P$ | = | \(\left(44, 242\right)\) | \(\left(596, 14249\right)\) |
$\hat{h}(P)$ | ≈ | $0.87798658204026519060156887146$ | $3.2731977487188494654612466037$ |
Torsion generators
\( \left(21, -11\right) \)
Integral points
\( \left(21, -11\right) \), \( \left(22, 22\right) \), \( \left(22, -45\right) \), \( \left(39, 193\right) \), \( \left(39, -233\right) \), \( \left(44, 242\right) \), \( \left(44, -287\right) \), \( \left(67, 495\right) \), \( \left(67, -563\right) \), \( \left(149, 1733\right) \), \( \left(149, -1883\right) \), \( \left(596, 14249\right) \), \( \left(596, -14846\right) \), \( \left(1079, 34903\right) \), \( \left(1079, -35983\right) \), \( \left(9037, 854577\right) \), \( \left(9037, -863615\right) \)
Invariants
Conductor: | \( 7406 \) | = | $2 \cdot 7 \cdot 23^{2}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $-4145004892 $ | = | $-1 \cdot 2^{2} \cdot 7 \cdot 23^{6} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( -\frac{15625}{28} \) | = | $-1 \cdot 2^{-2} \cdot 5^{6} \cdot 7^{-1}$ | 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: | $0.53566220943957695688029747309\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $-1.0320848985249978885230789428\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $1.0171207213859017\dots$ | |||
Szpiro ratio: | $3.3534894290024813\dots$ |
BSD invariants
Analytic rank: | $2$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $2.7901719696349524028944679173\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.55276805836035872247836995442\dots$ | 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: | $ 8 $ = $ 2\cdot1\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: | $2$ | 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^{(2)}(E,1)/2! $ ≈ $ 3.0846358842932208302860346039 $ | 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
$\displaystyle 3.084635884 \approx L^{(2)}(E,1)/2! \overset{?}{=} \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.552768 \cdot 2.790172 \cdot 8}{2^2} \approx 3.084635884$
Modular invariants
For more coefficients, see the Downloads section to the right.
Modular degree: | 4224 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | yes | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 3 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $2$ | $I_{2}$ | Non-split multiplicative | 1 | 1 | 2 | 2 |
$7$ | $1$ | $I_{1}$ | Non-split multiplicative | 1 | 1 | 1 | 1 |
$23$ | $4$ | $I_0^{*}$ | 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 |
---|---|---|
$2$ | 2B | 8.6.0.1 |
$3$ | 3B | 9.12.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 11592 = 2^{3} \cdot 3^{2} \cdot 7 \cdot 23 \), index $864$, genus $21$, and generators
$\left(\begin{array}{rr} 1 & 36 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 11557 & 36 \\ 11556 & 37 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 36 & 1 \end{array}\right),\left(\begin{array}{rr} 5797 & 9108 \\ 2530 & 9937 \end{array}\right),\left(\begin{array}{rr} 11248 & 8073 \\ 6095 & 11110 \end{array}\right),\left(\begin{array}{rr} 1007 & 0 \\ 0 & 11591 \end{array}\right),\left(\begin{array}{rr} 1289 & 9108 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 8695 & 9108 \\ 8326 & 9937 \end{array}\right),\left(\begin{array}{rr} 19 & 36 \\ 3240 & 6139 \end{array}\right),\left(\begin{array}{rr} 1 & 18 \\ 14 & 253 \end{array}\right)$.
The torsion field $K:=\Q(E[11592])$ is a degree-$3722877075456$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/11592\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3, 6, 9 and 18.
Its isogeny class 7406.a
consists of 6 curves linked by isogenies of
degrees dividing 18.
Twists
The minimal quadratic twist of this elliptic curve is 14.a5, its twist by $-23$.
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 |
---|---|---|---|
$2$ | \(\Q(\sqrt{-7}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | Not in database |
$2$ | \(\Q(\sqrt{-23}) \) | \(\Z/6\Z\) | Not in database |
$4$ | 4.2.236992.3 | \(\Z/4\Z\) | Not in database |
$4$ | \(\Q(\sqrt{-7}, \sqrt{-23})\) | \(\Z/2\Z \oplus \Z/6\Z\) | Not in database |
$6$ | 6.2.12620001744.2 | \(\Z/6\Z\) | Not in database |
$6$ | 6.0.29212967.1 | \(\Z/18\Z\) | Not in database |
$8$ | 8.0.33713166140416.20 | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$8$ | 8.0.2752095195136.29 | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$8$ | 8.0.56165208064.2 | \(\Z/12\Z\) | Not in database |
$12$ | deg 12 | \(\Z/3\Z \oplus \Z/6\Z\) | Not in database |
$12$ | 12.0.66332546446715136.1 | \(\Z/18\Z\) | Not in database |
$12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | Not in database |
$12$ | 12.0.41816474606211361.1 | \(\Z/2\Z \oplus \Z/18\Z\) | Not in database |
$16$ | deg 16 | \(\Z/8\Z\) | Not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | Not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | 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 | ord | ss | nonsplit | ss | ord | ord | ord | add | ord | ord | ord | ord | ord | ord |
$\lambda$-invariant(s) | 4 | 2 | 6,4 | 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 |
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.