Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+7125x+396250\)
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(homogenize, simplify) |
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\(y^2z=x^3+7125xz^2+396250z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+7125x+396250\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(125, 1800)$ | $0.20251161659504527109172310442$ | $\infty$ |
| $(29, 792)$ | $2.2138736835159214724272983437$ | $\infty$ |
Integral points
\((-25,\pm 450)\), \((-19,\pm 504)\), \((26,\pm 774)\), \((29,\pm 792)\), \((125,\pm 1800)\), \((150,\pm 2200)\), \((525,\pm 12200)\), \((575,\pm 13950)\), \((719,\pm 19422)\), \((25325,\pm 4030200)\), \((272429,\pm 142193592)\)
Invariants
| Conductor: | $N$ | = | \( 46800 \) | = | $2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 13$ |
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| Discriminant: | $\Delta$ | = | $-90979200000000$ | = | $-1 \cdot 2^{13} \cdot 3^{7} \cdot 5^{8} \cdot 13 $ |
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| j-invariant: | $j$ | = | \( \frac{34295}{78} \) | = | $2^{-1} \cdot 3^{-1} \cdot 5 \cdot 13^{-1} \cdot 19^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.3625575142229576154421742833$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.95285441896044278940652001210$ |
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| $abc$ quality: | $Q$ | ≈ | $0.8051698317586183$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.654832437605826$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 2$ |
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| Mordell-Weil rank: | $r$ | = | $ 2$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.43999905151900349909034510771$ |
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| Real period: | $\Omega$ | ≈ | $0.41937298844110248077687359981$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 48 $ = $ 2^{2}\cdot2^{2}\cdot3\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L^{(2)}(E,1)/2!$ | ≈ | $8.8571384230452052493029040358 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 8.857138423 \approx L^{(2)}(E,1)/2! & \overset{?}{=} \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.419373 \cdot 0.439999 \cdot 48}{1^2} \\ & \approx 8.857138423\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 161280 |
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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 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{5}^{*}$ | additive | -1 | 4 | 13 | 1 |
| $3$ | $4$ | $I_{1}^{*}$ | additive | -1 | 2 | 7 | 1 |
| $5$ | $3$ | $IV^{*}$ | additive | -1 | 2 | 8 | 0 |
| $13$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
Galois representations
The $\ell$-adic Galois representation has maximal image for all primes $\ell$.
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 312 = 2^{3} \cdot 3 \cdot 13 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 145 & 2 \\ 145 & 3 \end{array}\right),\left(\begin{array}{rr} 311 & 2 \\ 310 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 311 & 0 \end{array}\right),\left(\begin{array}{rr} 79 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 209 & 2 \\ 209 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 157 & 2 \\ 157 & 3 \end{array}\right)$.
The torsion field $K:=\Q(E[312])$ is a degree-$966131712$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/312\Z)$.
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 |
|---|---|---|---|
| $2$ | additive | $4$ | \( 2925 = 3^{2} \cdot 5^{2} \cdot 13 \) |
| $3$ | additive | $8$ | \( 5200 = 2^{4} \cdot 5^{2} \cdot 13 \) |
| $5$ | additive | $14$ | \( 1872 = 2^{4} \cdot 3^{2} \cdot 13 \) |
| $13$ | nonsplit multiplicative | $14$ | \( 3600 = 2^{4} \cdot 3^{2} \cdot 5^{2} \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 46800fa consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 1950z1, its twist by $60$.
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 |
|---|---|---|---|
| $3$ | 3.1.7800.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.0.18982080000.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $8$ | deg 8 | \(\Z/3\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | 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 | add | add | ord | ord | nonsplit | ord | ord | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | - | - | - | 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 |
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.