Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2=x^3-1228483x-575017406\) | (homogenize, simplify) |
\(y^2z=x^3-1228483xz^2-575017406z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-1228483x-575017406\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z\)
Infinite order Mordell-Weil generators and heights
$P$ | = | \(\left(1577, 37544\right)\) | \(\left(1369, 17576\right)\) |
$\hat{h}(P)$ | ≈ | $1.0072379971455304178034490064$ | $1.6553877326521635106388169645$ |
Integral points
\((1369,\pm 17576)\), \((1577,\pm 37544)\), \((3735,\pm 216658)\), \((4697,\pm 311896)\), \((6270,\pm 488072)\), \((12887,\pm 1457326)\), \((67279,\pm 17448574)\)
Invariants
Conductor: | \( 75088 \) | = | $2^{4} \cdot 13 \cdot 19^{2}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $-24183269488299843584 $ | = | $-1 \cdot 2^{13} \cdot 13^{7} \cdot 19^{6} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( -\frac{1064019559329}{125497034} \) | = | $-1 \cdot 2^{-1} \cdot 3^{3} \cdot 13^{-7} \cdot 41^{3} \cdot 83^{3}$ | 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: | $2.4551611962465501046177529074\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $0.28979452610338456519600707000\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $1.0626891964834324\dots$ | |||
Szpiro ratio: | $4.797877354120879\dots$ |
BSD invariants
Analytic rank: | $2$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $1.6213691552529616488557034099\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.071229610687493235273444391117\dots$ | 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: | $ 56 $ = $ 2^{2}\cdot7\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: | $1$ | 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! $ ≈ $ 6.4674116477251811997812155383 $ | 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
$\displaystyle 6.467411648 \approx L^{(2)}(E,1)/2! \overset{?}{=} \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.071230 \cdot 1.621369 \cdot 56}{1^2} \approx 6.467411648$
Modular invariants
Modular form 75088.2.a.b
For more coefficients, see the Downloads section to the right.
Modular degree: | 2201472 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | no | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 3 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $4$ | $I_{5}^{*}$ | Additive | -1 | 4 | 13 | 1 |
$13$ | $7$ | $I_{7}$ | Split multiplicative | -1 | 1 | 7 | 7 |
$19$ | $2$ | $I_0^{*}$ | 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 |
---|---|---|
$7$ | 7B.6.3 | 7.24.0.2 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 13832 = 2^{3} \cdot 7 \cdot 13 \cdot 19 \), index $96$, genus $2$, and generators
$\left(\begin{array}{rr} 13819 & 14 \\ 13818 & 15 \end{array}\right),\left(\begin{array}{rr} 2183 & 0 \\ 0 & 13831 \end{array}\right),\left(\begin{array}{rr} 3457 & 10906 \\ 8911 & 7181 \end{array}\right),\left(\begin{array}{rr} 913 & 5624 \\ 3990 & 6385 \end{array}\right),\left(\begin{array}{rr} 6385 & 2926 \\ 4655 & 6651 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 14 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 14 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 8 & 5 \\ 91 & 57 \end{array}\right),\left(\begin{array}{rr} 6917 & 2926 \\ 8379 & 6651 \end{array}\right)$.
The torsion field $K:=\Q(E[13832])$ is a degree-$104081369333760$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/13832\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
7.
Its isogeny class 75088.b
consists of 2 curves linked by isogenies of
degree 7.
Twists
The minimal quadratic twist of this elliptic curve is 26.b1, its twist by $76$.
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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:
$[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
---|---|---|---|
$3$ | 3.1.104.1 | \(\Z/2\Z\) | Not in database |
$6$ | 6.0.1124864.1 | \(\Z/2\Z \oplus \Z/2\Z\) | Not in database |
$6$ | 6.0.7377869632.2 | \(\Z/7\Z\) | Not in database |
$8$ | deg 8 | \(\Z/3\Z\) | Not in database |
$12$ | deg 12 | \(\Z/4\Z\) | Not in database |
$14$ | 14.2.635693367356250458067828736.1 | \(\Z/7\Z\) | Not in database |
$18$ | 18.0.7939862701662327913375442615295910346752.1 | \(\Z/14\Z\) | 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 | add | ss | ord | ord | ord | split | ord | add | ord | ord | ord | ord | ss | ord | ord |
$\lambda$-invariant(s) | - | 2,2 | 4 | 2 | 2 | 3 | 2 | - | 2 | 2 | 2 | 2 | 2,2 | 2 | 2 |
$\mu$-invariant(s) | - | 0,0 | 0 | 1 | 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.