Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-2523x+56522\)
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(homogenize, simplify) |
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\(y^2z=x^3-2523xz^2+56522z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-2523x+56522\)
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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 |
|---|---|---|
| $(7, 198)$ | $0.65988684062113670543506180515$ | $\infty$ |
| $(29, 88)$ | $1.1206320921279702366312835591$ | $\infty$ |
| $(-59, 0)$ | $0$ | $2$ |
Integral points
\( \left(-59, 0\right) \), \((-58,\pm 88)\), \((-49,\pm 250)\), \((-11,\pm 288)\), \((7,\pm 198)\), \((22,\pm 108)\), \((29,\pm 88)\), \((31,\pm 90)\), \((61,\pm 360)\), \((106,\pm 990)\), \((133,\pm 1440)\), \((431,\pm 8890)\), \((733,\pm 19800)\), \((7861,\pm 696960)\)
Invariants
| Conductor: | $N$ | = | \( 102960 \) | = | $2^{4} \cdot 3^{2} \cdot 5 \cdot 11 \cdot 13$ |
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| Discriminant: | $\Delta$ | = | $-352271462400$ | = | $-1 \cdot 2^{12} \cdot 3^{7} \cdot 5^{2} \cdot 11^{2} \cdot 13 $ |
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| j-invariant: | $j$ | = | \( -\frac{594823321}{117975} \) | = | $-1 \cdot 3^{-1} \cdot 5^{-2} \cdot 11^{-2} \cdot 13^{-1} \cdot 29^{6}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.93859477082623364848068872046$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.30385855406776650663416601946$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9325722943536257$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.0677198028185426$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 2$ |
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| Mordell-Weil rank: | $r$ | = | $ 2$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.69620524822569715136508488872$ |
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| Real period: | $\Omega$ | ≈ | $0.91834616683334749444058548754$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 64 $ = $ 2^{2}\cdot2^{2}\cdot2\cdot2\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L^{(2)}(E,1)/2!$ | ≈ | $10.229718736597250893736287520 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 10.229718737 \approx L^{(2)}(E,1)/2! & \overset{?}{=} \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.918346 \cdot 0.696205 \cdot 64}{2^2} \\ & \approx 10.229718737\end{aligned}$$
Modular invariants
Modular form 102960.2.a.p
For more coefficients, see the Downloads section to the right.
| Modular degree: | 114688 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{4}^{*}$ | additive | -1 | 4 | 12 | 0 |
| $3$ | $4$ | $I_{1}^{*}$ | additive | -1 | 2 | 7 | 1 |
| $5$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $11$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $13$ | $1$ | $I_{1}$ | split 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 | 2.3.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 8580 = 2^{2} \cdot 3 \cdot 5 \cdot 11 \cdot 13 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 7801 & 4 \\ 7022 & 9 \end{array}\right),\left(\begin{array}{rr} 1717 & 4 \\ 3434 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 5722 & 1 \\ 5719 & 0 \end{array}\right),\left(\begin{array}{rr} 2149 & 6436 \\ 2144 & 6435 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 5282 & 1 \\ 5939 & 0 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 8577 & 4 \\ 8576 & 5 \end{array}\right)$.
The torsion field $K:=\Q(E[8580])$ is a degree-$63764692992000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/8580\Z)$.
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 | $2$ | \( 117 = 3^{2} \cdot 13 \) |
| $3$ | additive | $8$ | \( 11440 = 2^{4} \cdot 5 \cdot 11 \cdot 13 \) |
| $5$ | nonsplit multiplicative | $6$ | \( 20592 = 2^{4} \cdot 3^{2} \cdot 11 \cdot 13 \) |
| $11$ | nonsplit multiplicative | $12$ | \( 9360 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 13 \) |
| $13$ | split multiplicative | $14$ | \( 7920 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 11 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 102960.p
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 2145.f2, its twist by $12$.
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 |
| $4$ | 4.2.1887600.2 | \(\Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.5419374348960000.4 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | 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 | nonsplit | ord | nonsplit | split | ord | ord | ord | ss | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | - | - | 4 | 2 | 2 | 5 | 4 | 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 |
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.