Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3+x^2+3748475x-2711661875\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3+x^2z+3748475xz^2-2711661875z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+4858022925x-126588166787250\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(1375625/64, 1614286825/512)$ | $9.2070510867049385321221453897$ | $\infty$ |
| $(650, -325)$ | $0$ | $2$ |
Integral points
\( \left(650, -325\right) \)
Invariants
| Conductor: | $N$ | = | \( 124950 \) | = | $2 \cdot 3 \cdot 5^{2} \cdot 7^{2} \cdot 17$ |
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| Discriminant: | $\Delta$ | = | $-6551086651146240000000$ | = | $-1 \cdot 2^{24} \cdot 3 \cdot 5^{7} \cdot 7^{8} \cdot 17^{2} $ |
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| j-invariant: | $j$ | = | \( \frac{3168685387909439}{3563732336640} \) | = | $2^{-24} \cdot 3^{-1} \cdot 5^{-1} \cdot 7^{-2} \cdot 17^{-2} \cdot 191^{3} \cdot 769^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.8726121769948394977312571533$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.0949381462501326578782011150$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9594119735978286$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.859048246102883$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $9.2070510867049385321221453897$ |
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| Real period: | $\Omega$ | ≈ | $0.071991858124787392103142985258$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 32 $ = $ 2\cdot1\cdot2^{2}\cdot2\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $5.3026617246538521341886705360 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 5.302661725 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.071992 \cdot 9.207051 \cdot 32}{2^2} \\ & \approx 5.302661725\end{aligned}$$
Modular invariants
Modular form 124950.2.a.ce
For more coefficients, see the Downloads section to the right.
| Modular degree: | 10616832 |
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| $ \Gamma_0(N) $-optimal: | yes | |
| Manin constant: | 1 |
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Local data at primes of bad reduction
This elliptic curve is not semistable. There are 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{24}$ | nonsplit multiplicative | 1 | 1 | 24 | 24 |
| $3$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $5$ | $4$ | $I_{1}^{*}$ | additive | 1 | 2 | 7 | 1 |
| $7$ | $2$ | $I_{2}^{*}$ | additive | -1 | 2 | 8 | 2 |
| $17$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
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 | 16.24.0.13 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 28560 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \cdot 17 \), index $192$, genus $1$, and generators
$\left(\begin{array}{rr} 15373 & 8176 \\ 3584 & 24165 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 13616 & 20405 \\ 16275 & 24466 \end{array}\right),\left(\begin{array}{rr} 28545 & 16 \\ 28544 & 17 \end{array}\right),\left(\begin{array}{rr} 22961 & 8176 \\ 20090 & 5391 \end{array}\right),\left(\begin{array}{rr} 1 & 8176 \\ 7140 & 7141 \end{array}\right),\left(\begin{array}{rr} 7328 & 8155 \\ 12285 & 4094 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 15 & 2 \\ 28462 & 28547 \end{array}\right),\left(\begin{array}{rr} 8159 & 0 \\ 0 & 28559 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 28556 & 28557 \end{array}\right)$.
The torsion field $K:=\Q(E[28560])$ is a degree-$465740884869120$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/28560\Z)$.
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 | $4$ | \( 3675 = 3 \cdot 5^{2} \cdot 7^{2} \) |
| $3$ | nonsplit multiplicative | $4$ | \( 20825 = 5^{2} \cdot 7^{2} \cdot 17 \) |
| $5$ | additive | $18$ | \( 4998 = 2 \cdot 3 \cdot 7^{2} \cdot 17 \) |
| $7$ | additive | $32$ | \( 2550 = 2 \cdot 3 \cdot 5^{2} \cdot 17 \) |
| $17$ | split multiplicative | $18$ | \( 7350 = 2 \cdot 3 \cdot 5^{2} \cdot 7^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 4 and 8.
Its isogeny class 124950bc
consists of 6 curves linked by isogenies of
degrees dividing 8.
Twists
The minimal quadratic twist of this elliptic curve is 3570t1, its twist by $-35$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-15}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{21}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-35}) \) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-15}, \sqrt{21})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-35}, \sqrt{-255})\) | \(\Z/8\Z\) | not in database |
| $4$ | \(\Q(\sqrt{17}, \sqrt{-35})\) | \(\Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.4.7001316000000.6 | \(\Z/8\Z\) | not in database |
| $8$ | 8.0.10152029750625.1 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/6\Z\) | 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 | \(\Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/12\Z\) | 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 | nonsplit | add | add | ord | ord | split | ord | ss | ord | ord | ord | ord | ord | ss |
| $\lambda$-invariant(s) | 9 | 3 | - | - | 1 | 1 | 2 | 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 |
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.