Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+x^2-298184x+72762996\)
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(homogenize, simplify) |
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\(y^2z=x^3+x^2z-298184xz^2+72762996z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-24152931x+53116682850\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-290, 11616)$ | $1.4407333981138410635177352018$ | $\infty$ |
| $(436, 5082)$ | $2.4247822276321353591192501727$ | $\infty$ |
| $(-642, 0)$ | $0$ | $2$ |
Integral points
\( \left(-642, 0\right) \), \((-290,\pm 11616)\), \((-66,\pm 9600)\), \((382,\pm 3840)\), \((436,\pm 5082)\), \((583,\pm 9870)\), \((2020,\pm 87846)\)
Invariants
| Conductor: | $N$ | = | \( 75504 \) | = | $2^{4} \cdot 3 \cdot 11^{2} \cdot 13$ |
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| Discriminant: | $\Delta$ | = | $-596642105586548736$ | = | $-1 \cdot 2^{16} \cdot 3^{3} \cdot 11^{10} \cdot 13 $ |
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| j-invariant: | $j$ | = | \( -\frac{404075127457}{82223856} \) | = | $-1 \cdot 2^{-4} \cdot 3^{-3} \cdot 11^{-4} \cdot 13^{-1} \cdot 7393^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.1332134318600506477464446406$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.24111861490092006629824073016$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9803976849267315$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.427668578579052$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 2$ |
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| Mordell-Weil rank: | $r$ | = | $ 2$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $3.4266430928388789143591690814$ |
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| Real period: | $\Omega$ | ≈ | $0.27773033975184755102587951952$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 48 $ = $ 2^{2}\cdot3\cdot2^{2}\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L^{(2)}(E,1)/2!$ | ≈ | $11.420193004589562370266967475 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 11.420193005 \approx L^{(2)}(E,1)/2! & \overset{?}{=} \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.277730 \cdot 3.426643 \cdot 48}{2^2} \\ & \approx 11.420193005\end{aligned}$$
Modular invariants
Modular form 75504.2.a.bp
For more coefficients, see the Downloads section to the right.
| Modular degree: | 829440 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{8}^{*}$ | additive | -1 | 4 | 16 | 4 |
| $3$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
| $11$ | $4$ | $I_{4}^{*}$ | additive | -1 | 2 | 10 | 4 |
| $13$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
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 | 8.12.0.12 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 3432 = 2^{3} \cdot 3 \cdot 11 \cdot 13 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 7 & 6 \\ 3426 & 3427 \end{array}\right),\left(\begin{array}{rr} 3425 & 8 \\ 3424 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 3011 & 3006 \\ 434 & 2147 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 2495 & 3424 \\ 3116 & 3399 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 268 & 1 \\ 2399 & 6 \end{array}\right),\left(\begin{array}{rr} 1148 & 1 \\ 2311 & 6 \end{array}\right),\left(\begin{array}{rr} 2999 & 3000 \\ 1274 & 2993 \end{array}\right)$.
The torsion field $K:=\Q(E[3432])$ is a degree-$531372441600$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/3432\Z)$.
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 | $2$ | \( 4719 = 3 \cdot 11^{2} \cdot 13 \) |
| $3$ | split multiplicative | $4$ | \( 25168 = 2^{4} \cdot 11^{2} \cdot 13 \) |
| $11$ | additive | $72$ | \( 624 = 2^{4} \cdot 3 \cdot 13 \) |
| $13$ | nonsplit multiplicative | $14$ | \( 5808 = 2^{4} \cdot 3 \cdot 11^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 75504.bp
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 858.f4, its twist by $44$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-39}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{11}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-429}) \) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{11}, \sqrt{-39})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.824286838476816.2 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.4.1459418628096.5 | \(\Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/8\Z\) | 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 | \(\Z/2\Z \oplus \Z/8\Z\) | 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 | add | split | ord | ss | add | nonsplit | ord | ss | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | - | 3 | 4 | 2,2 | - | 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 |
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.