Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-x^2-2524419x+1541499925\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-x^2z-2524419xz^2+1541499925z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-40390707x+98615604494\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(254, 30149\right) \) | $1.5407434965515204687448972170$ | $\infty$ |
| \( \left(-1834, 917\right) \) | $0$ | $2$ |
| \( \left(950, -475\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([254:30149:1]\) | $1.5407434965515204687448972170$ | $\infty$ |
| \([-1834:917:1]\) | $0$ | $2$ |
| \([950:-475:1]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(1015, 242208\right) \) | $1.5407434965515204687448972170$ | $\infty$ |
| \( \left(-7337, 0\right) \) | $0$ | $2$ |
| \( \left(3799, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(-1834, 917\right) \), \( \left(254, 30149\right) \), \( \left(254, -30403\right) \), \( \left(689, 11009\right) \), \( \left(689, -11698\right) \), \( \left(950, -475\right) \), \( \left(975, 2030\right) \), \( \left(975, -3005\right) \), \( \left(1166, 12917\right) \), \( \left(1166, -14083\right) \), \( \left(1791, 51667\right) \), \( \left(1791, -53458\right) \)
\([-1834:917:1]\), \([254:30149:1]\), \([254:-30403:1]\), \([689:11009:1]\), \([689:-11698:1]\), \([950:-475:1]\), \([975:2030:1]\), \([975:-3005:1]\), \([1166:12917:1]\), \([1166:-14083:1]\), \([1791:51667:1]\), \([1791:-53458:1]\)
\( \left(-7337, 0\right) \), \((1015,\pm 242208)\), \((2755,\pm 90828)\), \( \left(3799, 0\right) \), \((3899,\pm 20140)\), \((4663,\pm 108000)\), \((7163,\pm 420500)\)
Invariants
| Conductor: | $N$ | = | \( 75690 \) | = | $2 \cdot 3^{2} \cdot 5 \cdot 29^{2}$ |
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| Minimal Discriminant: | $\Delta$ | = | $3902635809081000000$ | = | $2^{6} \cdot 3^{8} \cdot 5^{6} \cdot 29^{6} $ |
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| j-invariant: | $j$ | = | \( \frac{4102915888729}{9000000} \) | = | $2^{-6} \cdot 3^{-2} \cdot 5^{-6} \cdot 7^{3} \cdot 2287^{3}$ |
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| Endomorphism ring: | $\mathrm{End}(E)$ | = | $\Z$ | |||
| Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z\) (no potential complex multiplication) |
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| Sato-Tate group: | $\mathrm{ST}(E)$ | = | $\mathrm{SU}(2)$ | |||
| Faltings height: | $h_{\mathrm{Faltings}}$ | ≈ | $2.4505464542636708848698848955$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.21759239493637902558062626086$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0522127879225598$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.970284227833859$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $1.5407434965515204687448972170$ |
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| Real period: | $\Omega$ | ≈ | $0.24838534055940911822060812014$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 192 $ = $ 2\cdot2^{2}\cdot( 2 \cdot 3 )\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $4$ |
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| Special value: | $ L'(E,1)$ | ≈ | $4.5923771772677303994741996226 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 4.592377177 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.248385 \cdot 1.540743 \cdot 192}{4^2} \\ & \approx 4.592377177\end{aligned}$$
Modular invariants
Modular form 75690.2.a.m
For more coefficients, see the Downloads section to the right.
| Modular degree: | 2322432 |
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| $ \Gamma_0(N) $-optimal: | no | |
| Manin constant: | 1 |
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Local data at primes of bad reduction
This elliptic curve is not semistable. There are 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{6}$ | nonsplit multiplicative | 1 | 1 | 6 | 6 |
| $3$ | $4$ | $I_{2}^{*}$ | additive | -1 | 2 | 8 | 2 |
| $5$ | $6$ | $I_{6}$ | split multiplicative | -1 | 1 | 6 | 6 |
| $29$ | $4$ | $I_0^{*}$ | 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 | $\ell$-adic index |
|---|---|---|---|
| $2$ | 2Cs | 2.6.0.1 | $6$ |
| $3$ | 3B | 3.4.0.1 | $4$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 3480 = 2^{3} \cdot 3 \cdot 5 \cdot 29 \), index $384$, genus $5$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 3469 & 12 \\ 3468 & 13 \end{array}\right),\left(\begin{array}{rr} 697 & 3132 \\ 1914 & 1567 \end{array}\right),\left(\begin{array}{rr} 839 & 0 \\ 0 & 3479 \end{array}\right),\left(\begin{array}{rr} 3073 & 1914 \\ 986 & 1565 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1741 & 3132 \\ 1566 & 1393 \end{array}\right),\left(\begin{array}{rr} 697 & 1566 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 9 & 4 \\ 3464 & 3473 \end{array}\right)$.
The torsion field $K:=\Q(E[3480])$ is a degree-$62860492800$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/3480\Z)$.
The table below list all primes $\ell$ for which the Serre invariants associated to the mod-$\ell$ Galois representation are exceptional.
| $\ell$ | Reduction type | Serre weight | Serre conductor |
|---|---|---|---|
| $2$ | nonsplit multiplicative | $4$ | \( 7569 = 3^{2} \cdot 29^{2} \) |
| $3$ | additive | $8$ | \( 841 = 29^{2} \) |
| $5$ | split multiplicative | $6$ | \( 15138 = 2 \cdot 3^{2} \cdot 29^{2} \) |
| $29$ | additive | $422$ | \( 90 = 2 \cdot 3^{2} \cdot 5 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 75690t
consists of 8 curves linked by isogenies of
degrees dividing 12.
Twists
The minimal quadratic twist of this elliptic curve is 30a6, its twist by $-87$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{29}) \) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $4$ | \(\Q(\sqrt{10}, \sqrt{-174})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-15}, \sqrt{-29})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{6}, \sqrt{29})\) | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $6$ | 6.0.4320438183.2 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $8$ | 8.0.146661788160000.52 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $8$ | 8.0.9166361760000.1 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $12$ | deg 12 | \(\Z/6\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/24\Z\) | not in database |
| $18$ | 18.6.2986892708433999953178024981000000000000.1 | \(\Z/2\Z \oplus \Z/18\Z\) | 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 | add | split | ord | ss | ord | ord | ord | ss | add | ord | ord | ord | ord | ss |
| $\lambda$-invariant(s) | 3 | - | 2 | 1 | 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 |
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.